Universality of two-dimensional critical cellular automata
Béla Bollobás, Hugo Duminil-Copin, Robert Morris, Paul Smith
UUNIVERSALITY OF TWO-DIMENSIONAL CRITICALCELLULAR AUTOMATA
B´ELA BOLLOB ´AS, HUGO DUMINIL-COPIN, ROBERT MORRIS, AND PAUL SMITH
Abstract.
We study the class of monotone, two-state, deterministic cellular au-tomata, in which sites are activated (or ‘infected’) by certain configurations ofnearby infected sites. These models have close connections to statistical physics,and several specific examples have been extensively studied in recent years byboth mathematicians and physicists. This general setting was first studied onlyrecently, however, by Bollob´as, Smith and Uzzell, who showed that the family ofall such ‘bootstrap percolation’ models on Z can be naturally partitioned intothree classes, which they termed subcritical, critical and supercritical.In this paper we determine the order of the threshold for percolation (com-plete occupation) for every critical bootstrap percolation model in two dimen-sions. This ‘universality’ theorem includes as special cases results of Aizenmanand Lebowitz, Gravner and Griffeath, Mountford, and van Enter and Hulshof,significantly strengthens bounds of Bollob´as, Smith and Uzzell, and complementsrecent work of Balister, Bollob´as, Przykucki and Smith on subcritical models. Contents
1. Introduction 22. Outline of the proof 83. Voracity and quasi-stability 164. The upper bound for balanced families 215. The upper bound for unbalanced families 256. Approximately internally filled sets 297. The lower bound for balanced families 448. The lower bound for unbalanced families 459. Conjectures for higher dimensions 75References 76
Date : March 5, 2018.2010
Mathematics Subject Classification.
Primary 60K35; Secondary 60C05.
Key words and phrases. cellular automata, bootstrap percolation, universality, critical proba-bility, metastability.B.B. is partially supported by NSF grant DMS 1301614 and MULTIPLEX grant no. 317532,R.M. by CNPq (Proc. 303275/2013-8), by FAPERJ (Proc. 201.598/2014), and by the ERC StartingGrant 680275 MALIG, and P.S. by a CNPq bolsa PDJ. a r X i v : . [ m a t h . P R ] M a r B. BOLLOB ´AS, H. DUMINIL-COPIN, R. MORRIS, AND P.J. SMITH Introduction
An important and challenging problem in statistical physics, probability theoryand combinatorics is to understand the typical global behaviour of so-called ‘latticemodels’, including cellular automata, percolation models, and spin models. Al-though these models are defined in terms of local interactions between the sitesof the lattice, it is typically observed in simulations that, in fixed dimensions, themacroscopic behaviour of the models does not seem to depend on the precise natureof these local interactions. Indeed, since the breakthrough work of Kadanoff [29]and the development of the renormalization group framework by Wilson [38], thisphenomenon of universality has been widely predicted to occur throughout statisti-cal physics (see, for example, [21,30,33]). Despite this, it has been proved rigorouslyin only a small handful of cases. One example of a model for which universality ispartially understood is the Ising model, for which it was proved recently that thecritical exponents exist and are equal on a large class of planar graphs [17, 22].Cellular automata are interacting particle systems whose update rules are localand homogeneous. Since their introduction by von Neumann [37] almost 50 yearsago, many particular such systems have been investigated, but no general theoryhas been developed for their study, and for many simple examples surprisingly lit-tle is known. In this paper we develop such a general theory for monotone, two-dimensional cellular automata with random initial configurations, which may alsobe thought of as monotone versions of the Glauber dynamics of the Ising model witharbitrary local interactions. The study of this general class of models was initiatedonly recently by Bollob´as, Smith and Uzzell [13], although many special cases hadbeen studied earlier, beginning with the work of Chalupa, Leath and Reich [16] in1979. We refer to these models as bootstrap percolation , but we emphasize that theyare vastly more general than the family of models that usually bears this name.The class of models we study is defined as follows. Fix d ∈ N and let U = { X , . . . , X m } be an arbitrary finite collection of finite subsets of Z d \ { } . Wecall U the update family of the process, each X ∈ U an update rule , and the processitself U -bootstrap percolation . Let the lattice Λ be either Z d or Z dn (the d -dimensionaldiscrete torus). Now given a set A ⊂ Λ of initially infected sites, set A = A , anddefine for each t (cid:62) A t +1 = A t ∪ (cid:8) x ∈ Λ : x + X ⊂ A t for some X ∈ U (cid:9) . Thus, a site x becomes infected at time t + 1 if the translate by x of one of the setsof the update family is already entirely infected at time t , and infected sites remaininfected forever. The set (cid:83) t (cid:62) A t of eventually infected sites is the closure of A ,denoted by [ A ]. We say that A percolates if [ A ] = Λ.As mentioned above, this model was first introduced (in its full generality) onlyrecently, in [13], although various special cases were introduced and studied muchearlier by several different authors; see for example [16, 18, 19, 23, 24]. Indeed, the NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 3 general class of U -bootstrap percolation models is easily seen to include as specificexamples all previously studied bootstrap percolation models on lattice graphs. Forexample, the update family of the classical r -neighbour model on Z d , the mostwell-studied of all models [1, 3, 4, 14, 15, 27], consists of the (cid:0) dr (cid:1) r -subsets of the 2 d nearest neighbours of the origin. The r -neighbour models are themselves examplesof threshold models , which, in their full generality, consist of the r -element subsetsof an arbitrary finite set Y ⊂ Z d \ { } .We say that an update family U is symmetric if X ∈ U implies − X ∈ U , so inparticular a threshold model is symmetric if and only if the set Y is centrally sym-metric. With a single exception (the work of Mountford [32] on the Duarte model,which exhibits a weaker notion of symmetry), we are not aware of any results onnon-symmetric update families before the work of [13]. In the absence of symmetry,many of the previously developed techniques appear to break down, and overcomingthis obstacle is one of the main tasks of this paper.Motivated by applications to statistical physics, we shall study the global be-haviour of the U -bootstrap process acting on random initial sets. Specifically, let ussay that a set A ⊂ Λ is p -random if each of the sites of Λ is included in A indepen-dently with probability p . The key question is that of how likely it is that a randomset A percolates on the lattice Λ; in particular, one would like to know how large p must be before percolation becomes likely. The point at which this phase transitionoccurs is measured by the critical probability , p c (Λ , U ) := inf (cid:110) p : P p (cid:0) A percolates in U -bootstrap percolation on Λ (cid:1) (cid:62) / (cid:111) , where P p denotes the product probability measure on Λ with density p . For the r -neighbour model on Z dn , with d fixed and n → ∞ , a great deal is knownabout the critical probability. Up to a constant factor, the threshold was determinedby Aizenman and Lebowitz [1] for r = 2, by Cerf and Cirillo [14] for d = r = 3, andby Cerf and Manzo [15] for all remaining 2 (cid:54) r (cid:54) d . The constant factor was laterimproved to a 1 + o (1) factor by Holroyd [27] in the case d = 2, by Balogh, Bollob´asand Morris [4] for d = 3, and by Balogh, Bollob´as, Duminil-Copin and Morris [3] forall d (cid:62)
4. The r -neighbour model has also attracted attention on lattices with thedimension d tending to infinity (for example the hypercube) [5, 6], and on graphsother than lattices, including trees [7, 11] and random graphs [8, 28].For lattice models other than the r -neighbour model, considerably less is known.Among the exceptions are two-dimensional balanced, symmetric threshold modelswith star-neighbourhoods , for which the critical probability was determined up toa constant factor by Gravner and Griffeath [24], and asymptotically by Duminil-Copin and Holroyd [18]. Some results about the critical probabilities of a ratherlimited number of so-called unbalanced models are also known; these were proved Thus a p -random set is one chosen according to the distribution P p . These terms are defined below.
B. BOLLOB ´AS, H. DUMINIL-COPIN, R. MORRIS, AND P.J. SMITH by Mountford [32], the authors of the present paper [9], van Enter and Hulshof [36],Duminil-Copin and van Enter [19], all in two dimensions, and by van Enter andFey [35] in three dimensions.For the remainder of the paper, with the exception of a brief discussion of higherdimensions in Section 9, we restrict our attention to the case d = 2. As we shallsee shortly, one of the key properties of the U -bootstrap process is that its roughglobal behaviour depends only on the action of the process on discrete half-planes.In order to make this statement precise, let us introduce a little notation. For each u ∈ S , let H u := { x ∈ Z : (cid:104) x, u (cid:105) < } be the discrete half-plane whose boundary isperpendicular to u . We say that u is a stable direction if [ H u ] = H u and we denoteby S = S ( U ) ⊂ S the collection of stable directions.The following classification of two-dimensional update families was proposed byBollob´as, Smith and Uzzell [13]. Definition 1.1.
An update family U is: • subcritical if every semicircle in S has infinite intersection with S ; • critical if there exists a semicircle in S that has finite intersection with S ,and if every open semicircle in S has non-empty intersection with S ; • supercritical if there exists an open semicircle in S that is disjoint from S .The justification of the above definition was completed in two stages. First,in their original paper, Bollob´as, Smith and Uzzell [13] proved that the criticalprobabilities of supercritical families are polynomial, while those of critical familiesare polylogarithmic. Later, Balister, Bollob´as, Przykucki and Smith [2] proved thatthe critical probabilities of subcritical models are bounded away from zero. Thecombination of the results of [13] and [2] may be summarized as follows : • if U is subcritical then lim inf n →∞ p c ( Z n , U ) > • if U is critical then p c ( Z n , U ) = (log n ) − Θ(1) ; • if U is supercritical then p c ( Z n , U ) = n − Θ(1) .In this paper we significantly strengthen the bounds of [13] by determining thethreshold p c ( Z n , U ) up to a constant factor for every critical update family. Thisresult, which may be thought of as a universality statement for two-dimensionalcritical bootstrap percolation, was previously known only in the case of one very re-strictive subclass of critical models [1,24], namely the symmetric, balanced thresholdmodels, and just two other specific models [32, 36].The form of the threshold function depends on two properties of U : the ‘difficulty’of U , and whether or not U is ‘balanced’. In order to explain what these terms mean,first we need a quantitative measure of how hard it is to grow in each direction. Our asymptotic notation is mostly standard; however, for the avoidance of ambiguity, precisedefinitions are given in Section 2.5.
NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 5 ( a ) ( b ) Figure 1.
Two examples of critical models, paused during their evo-lution on Z . In ( a ), the Duarte model, an unbalanced model withdrift; in ( b ), a balanced critical model.Let Q ⊂ S denote the set of rational directions on the circle , and for each u ∈ Q , let (cid:96) + u be the subset of the line (cid:96) u := { x ∈ Z : (cid:104) x, u (cid:105) = 0 } consisting ofthe origin and the sites to the right of the origin as one looks in the direction of u .Similarly, let (cid:96) − u := ( (cid:96) u \ (cid:96) + u ) ∪ { } consist of the origin and the sites to the left ofthe origin. Note that the line (cid:96) u is infinite for every u ∈ Q . Definition 1.2.
Given u ∈ Q , the difficulty of u is α ( u ) := (cid:40) min (cid:8) α + ( u ) , α − ( u ) (cid:9) if α + ( u ) < ∞ and α − ( u ) < ∞∞ otherwise,where α + ( u ) (respectively α − ( u )) is defined to be the minimum (possibly infinite)cardinality of a set Z ⊂ Z such that [ H u ∪ Z ] contains infinitely many sites of (cid:96) + u (respectively (cid:96) − u ). It follows from simple properties of stable sets (see Section 2.4)that α ( u ) > u is a stable direction. Now let C denote the collectionof open semicircles of S . We define the difficulty of U to be α = α ( U ) := min C ∈C max u ∈ C α ( u ) . (1)In Section 2 we discuss why these definitions of the difficulty of a direction underthe action of U and of the difficulty of U itself are the natural ones. The finaldefinition we need is as follows. Definition 1.3.
A critical update family U is balanced if there exists a closed semi-circle C such that α ( u ) (cid:54) α for all u ∈ C . It is said to be unbalanced otherwise. That is, the set of all u ∈ S such that u has rational or infinite gradient with respect to thestandard basis vectors. B. BOLLOB ´AS, H. DUMINIL-COPIN, R. MORRIS, AND P.J. SMITH
The distinction between the open semicircles in the definition of α and the closedsemicircles in the definition of balanced is subtle but important. It turns out thatgrowth under the action of balanced critical families is completely two-dimensional,while growth under the action of unbalanced critical families is asymptotically one-dimensional. Despite this, the analysis of unbalanced families presents by far thegreater number of difficulties.The following theorem is the main result of this paper. Theorem 1.4.
Let U be a critical two-dimensional bootstrap percolation updatefamily.(i) If U is balanced, then p c (cid:0) Z n , U (cid:1) = Θ (cid:18) n (cid:19) /α . (ii) If U is unbalanced, then p c (cid:0) Z n , U (cid:1) = Θ (cid:18) (log log n ) log n (cid:19) /α . On the infinite lattice (where the critical probability is zero), one can state anessentially equivalent version of Theorem 1.4 in terms of the infection time of theorigin. To be precise, given A ⊂ Z , define τ = τ ( A, U ) := min (cid:8) t (cid:62) ∈ A t (cid:9) to be the time at which the origin is infected in the U -bootstrap process on Z with A = A . We write ‘with high probability’ to mean ‘with probability tending to 1’. For a proof of the following theorem, see the earlier arXiv version of this paper [10].
Theorem 1.5.
Let U be a critical two-dimensional bootstrap percolation updatefamily, and let A be a p -random subset of Z .(i) If U is balanced, then, with high probability as p → , p α log τ = Θ(1) . (ii) If U is unbalanced, then, with high probability as p → , p α (cid:18) log 1 p (cid:19) − log τ = Θ(1) . We noted earlier that various special cases of Theorem 1.4 have already beenproved in the literature. The critical probability of the 2-neighbour model wasestablished by Aizenman and Lebowitz [1] using methods that have become centralto the study of bootstrap percolation, including the ‘rectangles process’ and thenotion of a ‘critical droplet’ (see Section 2 for details). Mountford [32] determined Moreover, we say that Z = Θ (cid:0) f ( p ) (cid:1) with high probability, where Z is a random variable, ifthere exist constants c, C > p ) such that cf ( p ) (cid:54) Z (cid:54) Cf ( p ) with highprobability. NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 7 the critical probability of the Duarte model, which is the unbalanced threshold modelconsisting of all two-element subsets of (cid:8) ( − , , (0 , , (0 , − (cid:9) . His proof was based on martingale techniques, which makes it unique among proofsof this type of theorem. Gravner and Griffeath [24] generalized the result of Aizen-man and Lebowitz to a class of balanced, symmetric threshold models, using some-what non-rigorous methods. The critical probability of one further unbalancedmodel, namely the one consisting of all three-element subsets of (cid:8) ( − , , ( − , , (0 , , (0 , − , (1 , , (2 , (cid:9) , was determined by van Enter and Hulshof [36], correcting an assertion of Gravnerand Griffeath [24]. Until now, the models studied by Mountford [32] and by vanEnter and Hulshof [36] were the only two unbalanced models whose critical proba-bilities were known, and they were, respectively, the unique such examples of ‘drift’and ‘non-drift’ unbalanced models. One property that all of these previously studied models share, and one thatsimplifies the problem enormously, is that of symmetry. In all but the Duarte model,the symmetry is particularly strong, in that X ∈ U if and only if − X ∈ U . Thesymmetry of the Duarte model is weaker (the useful property is that there existsa parallelogram of stable directions { u, − u, v, − v } ⊂ S ), but it is enough to makea significant difference to the proof. An important aspect of the general modelsthat we study – perhaps the most important aspect – is the lack of any symmetryassumptions. Indeed, it is little exaggeration to say that the main task of this paper(as was that of [13]) is to handle the lack of symmetry, which causes all previouslyknown techniques to break down.In all of the above cases (namely, the 2-neighbour model of Aizenman and Leb-owitz, the symmetric, balanced threshold models of Gravner and Griffeath, theDuarte model of Mountford, and the unbalanced model of van Enter and Hulshof)– but in no other fundamentally different cases – the critical probability has nowbeen determined up to a 1 + o (1) factor. These results are due to Holroyd [27],Duminil-Copin and Holroyd [18], the authors of the present paper [9] and Duminil-Copin and van Enter [19], respectively, and in some cases, even sharper results areknown [20, 25, 31]. Obtaining similarly sharp bounds for the general model is likelyto be an important, but extremely difficult, direction for future research.The organization of the rest of the paper is as follows. In Section 2 we give anoutline of the proof, we introduce some notation, and we recall a number of basicfacts about U -bootstrap percolation from [13]. In Section 3 we lay the ground-work for the proofs of the upper bounds of Theorem 1.4, which are then proved in These terms are explained in Section 2, but roughly speaking, the term ‘drift’ refers to thephenomenon that occurs when u ∈ S is such that α ( u ) = ∞ but min (cid:8) α − ( u ) , α + ( u ) (cid:9) < ∞ , whichin certain cases causes the growth to be biased in one direction. B. BOLLOB ´AS, H. DUMINIL-COPIN, R. MORRIS, AND P.J. SMITH
Sections 4 (balanced case) and 5 (unbalanced case). In Section 6 we define threedifferent notions of ‘approximately internally filled’ sets and prove a number of de-terministic properties of such sets. In Section 7 we deduce the lower bound in thebalanced case. The hardest part of the proof is the lower bound in the unbalancedcase, which is contained in Section 8. Finally, we end the paper with some openproblems, including a discussion of the problem in higher dimensions.2.
Outline of the proof
Let us begin by explaining why α ( u ) is the right definition of the difficulty ofgrowing in a direction u ∈ S . The key fact is that there is a sense (which isformalized in Lemma 3.4) in which α ( u ) measures how hard it is to infect an entirenew line in direction u , rather than merely an infinite subset of the line. Morespecifically, while the definition of α ( u ) only guarantees that there exist sets of α ( u )sites that will infect infinitely many new sites on the line (cid:96) u (with the help of H u ),one can show that only boundedly many copies of this set are needed to infect thewhole line. (This is false without the condition that both α − ( u ) and α + ( u ) arefinite.)Next let us see why the quantity α = α ( U ) defined in (1) is the constant oneshould expect to see in the exponent of the critical probability in Theorem 1.4. Inorder to do this, we need the definition of a droplet, which is just a polygon in Z .Droplets will be our means of controlling the growth of a set of infected sites. Definition 2.1.
Let
T ⊂ Q be finite. A T -droplet is a non-empty set of the form D = (cid:92) u ∈T (cid:0) H u + a u (cid:1) for some collection { a u ∈ Z : u ∈ T } .Reinterpreted in terms of droplets, the definition of α in (1) is equivalent to thestatement that there exist finite T -droplets for some set T ⊂ S such that α ( u ) (cid:62) α for all u ∈ T , but that the same is not true if α is replaced by any larger quantity.In other words, any finite set of infected sites is contained in a closed droplet suchthat a ‘cluster’ of at least α sites is needed to create non-localized new infections.In the other direction, the condition that there exists an open semicircle C ⊂ S such that every u ∈ C has difficulty at most α , which is implied by the definition of α , means that there is an interval of directions having difficulty at most α just largeenough for there to exist infinite sequences of nested droplets such that it is possibleto grow between consecutive droplets using only sets of α sites. (Note that in thegeneral model, unlike in symmetric bootstrap models, droplets do not necessarilygrow in all directions.)Before continuing with the outline of the proof, let us record two conventions thatwe use throughout the paper. First, U will always denote a fixed critical updatefamily, unless explicitly stated otherwise. Using results from Section 2.4, this is NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 9 equivalent to the statement that 1 (cid:54) α < ∞ . Second, A will always denote a p -random subset of either Z or Z n . We emphasize that, since we will usually beworking with droplets on a scale much smaller than n , most of the time we will nothave to worry about the difference between these two settings.2.1. Upper bounds.
The overall approach of the proofs of the upper bounds mir-rors that of previous works (see for example [1, 24, 36]). First we obtain a lowerbound of the form exp (cid:0) − O ( p − α ) (cid:1) for the probability that a droplet at a particu-lar intermediate scale (which is roughly p − Θ(1) ) is (almost) internally filled, where‘internally filled’ is defined as follows.
Definition 2.2.
A set X ⊂ Z is internally filled by A if X ⊂ [ X ∩ A ]. The eventthat X is internally filled by A is denoted I ( X ).We remark that in the older bootstrap percolation literature this event was re-ferred to as X being ‘internally spanned’ by A . However, following [3, 4], we willreserve that term for a different notion (see Definition 2.4).As alluded to before Definition 2.2, we will in fact usually show that certaindroplets are not quite exactly internally filled, but almost internally filled, wherewe use this terminology informally to mean that sites within distance O (1) may beused to help fill the droplet. The resulting loss of independence is not a problem,because the events are increasing and we bound them using Harris’s inequality.The key step in the proof is a bound of the form P p (cid:16) D m ⊂ (cid:2) D m − ∪ ( D m +1 ∩ A ) (cid:3)(cid:17) (cid:62) (cid:0) − (1 − p α ) Ω( m ) (cid:1) O (1) , where D ⊂ D ⊂ · · · is a certain sequence of nested droplets. This bound corre-sponds to the intuition that it is enough to find somewhere along each side of thedroplet a bounded number of sets of α sites contained in A . Once we have thisbound, we then deduce that with high probability there exists an internally filledcopy of this intermediate droplet in Z n , and that with high probability this dropletgrows to infect the whole torus.All of what we have just said assumes to some extent that the family is balanced.If it is unbalanced then the droplets in the nested sequence ( D m ) ∞ m =0 are somewhatdifferent: the sides (in the directions of growth) cannot grow linearly with m (as inthe balanced case), but instead all have the same length, and as a consequence thedroplets are much less ‘regular’ (for example, the initial droplet has width λ andheight λp − α log(1 /p ), where λ is a large constant, while in the balanced case it hasconstant size). The growth also features an extra step, in which an extremely longrectangular droplet grows a triangle of infected sites on its side.Two key deterministic properties of the growth process are needed to make theabove ideas work, for both balanced and unbalanced families. The first we havealready discussed: the statement that a bounded number of sets of α sites areenough to infect an entire new line; we refer to this principle as ‘voracity’ (see Section 3.1). The second is the ability to grow to the corners of droplets, not justto within a bounded distance of the corners. This is in general not possible with T a subset of the stable set S . However, using the idea of ‘quasi-stability’ introducedin [13], one can show that it can be done if a certain set of unstable directions isincluded in T . The details are given in Section 3.2.2.2. The lower bound for balanced families.
The lower bound for balancedupdate families is also not too difficult, but again requires refined versions of argu-ments from [13]. In order to sketch the proof, let us first briefly recall the argumentof Aizenman and Lebowitz [1] for the 2-neighbour model. Their key lemma statesthat if A ⊂ Z n percolates, then for every 1 (cid:54) k (cid:54) n , there exists an internally filledrectangle of semi-perimeter between k and 2 k . Using the simple and well-knownextremal result that such an internally filled rectangle contains at least k initiallyinfected sites, the bound follows from a straightforward calculation.The key lemma of Aizenman and Lebowitz is proved via the so-called ‘rectanglesprocess’, which is an algorithm for determining the exact closure of a finite set underthe 2-neighbour process. The algorithm proceeds by breaking down the bootstrapprocess into steps, each of which corresponds to the joining of two nearby rectanglesinto a larger rectangle. (Note that rectangles are closed under the 2-neighbourprocess.) One significant obstacle in the analysis of the general model is the lack ofa corresponding exact algorithm. Our solution is to use a process analogous to therectangles process but rather more complicated. This process is an adaptation ofthe ‘covering algorithm’ of Bollob´as, Smith and Uzzell [13], and we use it in orderto prove lemmas corresponding to those of [1]. Roughly speaking, we shall treatclusters of α nearby sites as seeds, cover each with a small S -droplet, and combinethem pairwise into larger droplets if they are sufficiently close to interact in the U -bootstrap process. The crucial deterministic property of the covering algorithmis that the remaining infections (those not in α -clusters) contribute a negligibleamount to the set of eventually infected sites; this is proved in Lemma 6.11.2.3. The lower bound for unbalanced families.
The proofs of the previous threeparts of the theorem are essentially refinements of established techniques. For thisfinal part of the theorem, however, these techniques do not seem to be useful, andinstead we introduce several substantial new ideas, including iterated hierarchies,the u -norm, and icebergs (see below). We mention these ideas only briefly in thissection, focusing instead on the broad structure of the proof, and on some of themost important definitions. A much fuller outline of the proof is given at the startof Section 8 (see also Section 6).The first observation we make (see Lemmas 2.9 and 6.2) is that there exist oppositestable directions u ∗ and − u ∗ that both have difficulty at least α + 1. We set S U = (cid:8) u ∗ , − u ∗ , u l , u r (cid:9) , NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 11 where u l and u r are stable directions on different sides of u ∗ , each of difficulty atleast α , and we consider only S U -droplets. Let us rotate our perspective so that u ∗ is vertical, and write h ( D ) and w ( D ) for the height and width of an S U -dropletrespectively.As in the balanced case, first we need an approximate rectangles process, whichwill allow us to say that if a large droplet is internally filled then it must containdroplets at all scales that are ‘approximately internally filled’. The covering algo-rithm is no longer useful to us because it is too crude to capture the biased nature ofthe geometry of unbalanced models. Instead we use a second algorithm, the ‘span-ning algorithm’, which is an adaptation of an idea introduced by Cerf and Cirilloin [14] and subsequently developed in [3, 4, 15, 18]. The algorithm uses the followingnotion of connectedness and the subsequent notion of being ‘internally spanned’,which is an approximation to being internally filled. Definition 2.3.
Let κ be a sufficiently large constant, to be defined explicitly in (14).Define a graph G κ with vertex set Z and edge set E ( G κ ) = (cid:8) xy : (cid:107) x − y (cid:107) (cid:54) κ (cid:9) .We say that a set S ⊂ Z is strongly connected if it is connected in the graph G κ . Definition 2.4.
Let
T ⊂ S be finite. A T -droplet D is internally spanned by A ifthere is a strongly connected set L ⊂ [ D ∩ A ] such that D is the smallest T -dropletcontaining L . We will write I × ( D ) for the event that D is internally spanned. As noted above, many previous authors have used the term ‘internally spanned’ tomean (what we refer to as) ‘internally filled’. We reemphasize that our terminology(which follows [3, 4], and seems to us more natural) is different.The spanning algorithm allows us to break down the formation of an internallyspanned droplet into intermediate steps in the same way that the original rectanglesprocess allowed Aizenman and Lebowitz [1] to break down the formation of aninternally filled droplet. Using the spanning algorithm we are able to say that if alarge droplet is internally spanned, then it contains internally spanned droplets atall smaller scales. The scale we are particularly interested in is the ‘critical’ scale,which for unbalanced models has the following specific meaning.
Definition 2.5.
Let U be unbalanced and let ξ > S U -droplet D is critical if either of the following conditions holds:( T ) w ( D ) (cid:54) p − α − / and ξp α log p (cid:54) h ( D ) (cid:54) ξp α log p ;( L ) p − α − / (cid:54) w ( D ) (cid:54) p − α − / and h ( D ) (cid:54) ξp α log p .Why might this be the right definition? It is certainly not surprising that thedroplet should be long and thin; this is the nature of unbalanced growth, as suggestedby the proof of the upper bound in Section 5. The height h = ξp α log p is such that Note that the event I × ( D ) also depends on T . However, we will only use this notation when T = S U , and so we trust this will therefore not cause the reader any confusion. an initial rectangle of height h and constant width will fail to grow sideways (thatis, perpendicular to u ∗ ) by a constant distance with probability roughly p O ( ξ ) , andtherefore one would expect the rectangle to grow sideways only to distance p − O ( ξ ) .The width w = p − α − / is such that the probability the rectangle grows to distance w is sufficiently small to compensate for the number of choices for the initial rectangle.The reason for there being two types of critical droplet is that the spanning algorithmcannot control the width and the height of the critical droplet simultaneously.In order to bound the probability that a critical droplet D is internally spanned, weshall show that, if the droplet is of type ( T ), then it is unlikely that [ D ∩ A ] containsa strongly connected set joining the ( − u ∗ )-side of D to the u ∗ -side, while if it is oftype ( L ), then instead it is unlikely that [ D ∩ A ] contains a strongly connected setjoining the u l -side to the u r -side. (The u -side of a droplet is defined precisely below.)These events are called ‘vertical crossings’ and ‘horizontal crossings’ respectively.There are several complications that occur while bounding the probabilities ofsuch crossings. Consider first vertical crossings, and note that, since α ( u ∗ ) (cid:62) α + 1,we have either min (cid:8) α + ( u ∗ ) , α − ( u ∗ ) (cid:9) (cid:62) α + 1, ormax (cid:8) α + ( u ∗ ) , α − ( u ∗ ) (cid:9) = ∞ and min (cid:8) α + ( u ∗ ) , α − ( u ∗ ) (cid:9) (cid:62) , (2)and similarly for − u ∗ . Since the former case is much easier to handle, let us assumein this sketch that (2) holds. (In this case we say the model exhibits drift .)For concreteness, suppose that α − ( u ∗ ) = ∞ and α + ( u ∗ ) = 1. Since we have apair { u ∗ , − u ∗ } of opposite stable directions, we may partition the droplet D intomany smaller sub-droplets of the same width, and bound the probability that eachis vertically crossed (possibly with help from above and below) independently, sincethese events depend on disjoint sets of infected sites. In order to bound these crossingprobabilities, we need several new ideas. First, we need a method of controllingthe range of the U -bootstrap process assisted by a half-plane. We achieve this byintroducing (in Section 6.3) a third algorithm for approximating the closure of a setof sites, which we call the ‘ u -iceberg algorithm’. Second, we need a close-to-best-possible bound for the probability that certain smaller sub-droplets are internallyspanned (following [4], we call these sub-droplets ‘savers’). In order to obtain such abound, we induct on the size of the droplet being crossed. (This means the proof forvertical crossings at a given scale depends on us having obtained sufficiently strongbounds for both vertical and horizontal crossings at the scale below.) Finally, weneed to deal with the ‘stretched geometry’ of drift models; we do so by introducinga family of norms (the ‘ u -norms’) that compress this geometry until it resemblesEuclidean space, and we also introduce a new concept of (‘weak’) connectedness;see Sections 8.2 and 8.3 respectively for the details.Now consider horizontal crossings, and observe that we no longer have symmetry,since − u l and − u r are in general not stable directions. This prevents us from parti-tioning into sub-droplets as with vertical crossings, and so to overcome this we usethe ‘hierarchy method’, which was introduced by Holroyd in [27] and subsequently NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 13 developed in [3, 4, 19, 25]. We would like to emphasize that the reason for our useof hierarchies is different to that of all previous works: here, the reason is the lackof symmetry between u l and u r (which is also why we do not need them for verti-cal crossings); previously the reason has been to prove sharp thresholds for criticalprobabilities in symmetric settings.In order to use hierarchies, we need three further ingredients: a bound on theprobability that ‘seeds’ (which are small sub-droplets) are internally spanned; abound on the probabilities of certain ( p times shorter) horizontal crossing events; anda bound on the number of hierarchies with a given number of ‘big seeds’. For thesewe use the induction hypothesis (once again, we need sufficiently strong bounds forboth vertical and horizontal crossings at the scale below), and the method describedabove for vertical crossings (although the details are somewhat simpler in this case).Since our use of induction on the size of the droplet amounts to iterating the aboveargument α times, we refer to this as the ‘method of iterated hierarchies’.2.4. Basic facts about U -bootstrap percolation. The U -bootstrap process ex-hibits a number of particularly simple and elegant properties, some of which we nowrecall from [13]. We begin with a description of the stable set S . We write [ v, w ] forthe closed interval of directions between v and w taken anticlockwise starting from v , and we say that [ v, w ] is rational if { v, w } ⊂ Q . Lemma 2.6 (Theorem 1.10 of [13]) . The stable set S is a finite union of rationalclosed intervals of S . The converse to Lemma 2.6 is also true (and is part of Theorem 1.10 of [13]): if
S ⊂ S is any set consisting of a finite union of rational closed intervals, then thereexists an update family U such that S = S ( U ). We shall not use this converse.The following simple properties of directions of infinite difficulty, which wereproved in [13], will also be useful. For completeness, we sketch the proofs. Lemma 2.7.
Let [ v, w ] ⊂ S with v (cid:54) = w be a connected component of S , and let u ∈ [ v, w ] ∩ Q . Then α + ( u ) < ∞ ⇔ u = v and α − ( u ) < ∞ ⇔ u = w. In particular, if u ∈ S ∩ Q , then α ( u ) < ∞ if and only u is an isolated point of S .Proof. We will show first that α + ( v ) < ∞ . To do so, observe that there exist unsta-ble directions arbitrarily close to (and to the right of) v . Choose such a direction v (cid:48) sufficiently close to v , and choose X ∈ U such that X ⊂ H v (cid:48) . Since the elements of X all lie within a finite distance of the origin, and v (cid:48) was chosen sufficiently closeto v , it follows that X ⊂ H v ∪ (cid:96) − v . Now, if Z is a set of consecutive sites of (cid:96) v thatcontains X \ H v , then [ H v ∪ Z ] ∩ (cid:96) + v is infinite, as required.In order to prove that α + ( u ) = ∞ for every u ∈ ( v, w ] ∩ Q , we will first showthat (for each such u ) there exists u (cid:48) ∈ ( v, u ) such that H u ∪ H u (cid:48) is closed under the U -bootstrap process. To do so, simply choose u (cid:48) closer to u than any u (cid:48)(cid:48) ∈ S \ { u } perpendicular to a vector in the set (cid:110) x − y : x, y ∈ (cid:91) X ∈U X ∪ { } , x (cid:54) = y (cid:111) . That we can do this follows easily from the fact that U is a finite collection of finitesets. Now, suppose that there exists a rule X ∈ U such that X ⊂ H u ∪ H u (cid:48) . Since u, u (cid:48) ∈ S , there must exist x, y ∈ X with x (cid:54)∈ H u and y (cid:54)∈ H u (cid:48) . But now x − y isperpendicular to a vector in the interval ( u (cid:48) , u ), which contradicts our choice of u (cid:48) .Next, let u ∈ ( v, w ] ∩ Q , and choose u (cid:48) ∈ ( v, u ) such that H u ∪ H u (cid:48) is closed.Now, for any finite set Z ⊂ Z , there exists y ∈ (cid:96) u such that y + Z ⊂ H u (cid:48) , andtherefore [ H u ∪ ( y + Z )] ⊂ H u ∪ H u (cid:48) . It follows that [ H u ∪ Z ] ∩ (cid:96) + u is finite, andhence α + ( u ) = ∞ , as required. The remaining claims now follow by symmetry, orare immediate from the definitions. (cid:3) Let us note, for emphasis, that the proof above also implies the following lemma.
Lemma 2.8. If u ∈ Q and α − ( u ) < ∞ , then there exists X ∈ U such that X ⊂ H u ∪ (cid:96) + u , and hence (cid:96) u ⊂ [ H u ∪ (cid:96) + u ] .Proof. By Lemma 2.7, there exist unstable directions arbitrarily close to (and tothe left of) u . Choose such a direction v sufficiently close to u , and choose X ∈ U such that X ⊂ H v . Since the elements of X all lie within a bounded distance of theorigin, it follows that X ⊂ H u ∪ (cid:96) + u , as required. (cid:3) We are now in a position to deduce the existence of opposite stable directions u ∗ and − u ∗ claimed earlier for unbalanced families U . Lemma 2.9.
Let U be an unbalanced critical update family. Then there exists u ∗ ∈ Q such that min (cid:8) α ( u ∗ ) , α ( − u ∗ ) (cid:9) (cid:62) α + 1 . Proof.
By the definition of α , there exists an open semicircle C ∈ C such that α ( u ) (cid:54) α for every u ∈ C . Moreover, since U is critical we have α < ∞ . Thus, ifone of the endpoints of C has difficulty at most α , then it is an isolated point of S ,by Lemma 2.6. Hence, rotating C slightly, we obtain a closed semicircle C (cid:48) such that α ( u ) (cid:54) α for all u ∈ C (cid:48) . But this contradicts our assumption that U is unbalanced,hence both endpoints of C have difficulty at least α + 1, as required. (cid:3) One final simple but important fact is that if u is not stable then H u grows to fillthe whole of Z . Lemma 2.10 (Lemma 3.1 of [13]) . If u (cid:54)∈ S , then [ H u ] = Z . Thus for every u ∈ S we have the dichotomy [ H u ] ∈ (cid:8) H u , Z } .2.5. Definitions and notation.
In this subsection we collect for ease of referencevarious conventions, definitions and notation that we shall use throughout the paper.
NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 15
Constants, and asymptotic notation.
All constants, including those impliedby the notation O ( · ), Ω( · ) and Θ( · ), are quantities that may depend on U (andother quantities where explicitly stated) but not on p . The parameter p will alwaysbe assumed to be sufficiently small relative to all other quantities. Our asymptoticnotation is mostly standard, although we just remark that if f and g are positivereal-valued functions of p , then we write f ( p ) = Ω (cid:0) g ( p ) (cid:1) if g ( p ) = O (cid:0) f ( p ) (cid:1) , and wewrite f ( p ) = Θ (cid:0) g ( p ) (cid:1) if both f ( p ) = O (cid:0) g ( p ) (cid:1) and g ( p ) = O (cid:0) f ( p ) (cid:1) . Furthermore,if c and c are constants, then c (cid:29) c (cid:29) c is sufficiently large,and c is sufficiently large depending on c , and 1 (cid:29) c (cid:29) c > c issufficiently small, and c is sufficiently small depending on c . (This last piece ofnotation is somewhat non-standard; we trust it will not cause any confusion.)2.5.2. Measuring sizes and distances.
The unadorned norm (cid:107) · (cid:107) always denotes theEuclidean norm on R , and (cid:104)· , ·(cid:105) always denotes the Euclidean inner product. Asremarked above, in Section 6 we will define a family of norms on R called the‘ u -norms’, which will be signified with a subscript u thus: (cid:107) · (cid:107) u .Now, for u ∈ S and a finite set K ⊂ Z , define the u -projection of K , π ( K, u ) := max (cid:8) (cid:104) x − y, u (cid:105) : x, y ∈ K (cid:9) . (3)Also, letdiam( K ) := max (cid:8) π ( K, u ) : u ∈ S (cid:9) = max (cid:8) (cid:107) x − y (cid:107) : x, y ∈ K (cid:9) be the diameter of K . Owing to the biased nature of the geometry, in the unbalancedsetting the diameter is usually not a useful measure of the size of K . Instead, wework with the height h ( K ) := π ( K, u ∗ ) = max (cid:8) (cid:104) x − y, u ∗ (cid:105) : x, y ∈ K (cid:9) , and the width w ( K ) := π ( K, u ⊥ ) = max (cid:8) (cid:104) x − y, u ⊥ (cid:105) : x, y ∈ K (cid:9) , where { u ∗ , − u ∗ } ⊂ S is the pair of opposite stable directions with difficulty strictlygreater than α given by Lemma 2.9, and u ⊥ ∈ S is either of the two unit vectorsthat are orthogonal to u ∗ . We will also make frequent use of the following constant,which we think of as being the ‘diameter’ of U : ν := max (cid:110) (cid:107) x − y (cid:107) : x, y ∈ X ∪ { } , X ∈ U (cid:111) . (4)We will define another constant ρ , which captures a different aspect of the “range”of the U -bootstrap process, in Section 6.Occasionally we shall want to talk about the distance between a site and a set ofsites, or between two sets of sites. We use the following standard conventions: (cid:107) x − Y (cid:107) := min (cid:8) (cid:107) x − y (cid:107) : y ∈ Y (cid:9) , and (cid:107) X − Y (cid:107) := min (cid:8) (cid:107) x − y (cid:107) : x ∈ X, y ∈ Y (cid:9) , whenever X and Y are finite subsets of Z . We also use analogous conventions forother measures of distance, such as the ‘ u -norms’ and inner products.2.5.3. Subsets of the plane. If u ∈ Q , then the collection of non-empty discretelines (cid:110)(cid:8) x ∈ Z : (cid:104) x − a, u (cid:105) = 0 (cid:9) : a ∈ Z (cid:111) is a discrete set, naturally indexed by Z . Thus, we may set (cid:96) u (0) := (cid:96) u and (for each j ∈ Z ) let (cid:96) u ( j ) denote the j th non-empty discrete line in the direction of u .For each u ∈ S and a ∈ R , we define the discrete half-planes H u ( a ) := (cid:8) x ∈ Z : (cid:104) x − a, u (cid:105) < (cid:9) . If a ∈ Z then we have H u ( a ) = H u + a , but this is false otherwise (since H u ⊂ Z ).Recall (cf. Definition 2.1) that a T -droplet is a non-empty set of the form D = (cid:92) u ∈T H u ( a u )for some collection { a u ∈ R : u ∈ T } . For each u ∈ T , the u -side of a T -droplet D is defined to be the set ∂ ( D, u ) := D ∩ (cid:96) u ( i ) , (5)where i is maximal subject to the set being non-empty. Finally, note that we canconsider droplets (even those with diameter larger than n ) as subsets of Z n by takingall x = ( x , x ) ∈ Z n such that ( x + in, x + jn ) ∈ D for some i, j ∈ Z .2.6. Probabilistic lemmas.
We end the section by recalling the correlation in-equalities of Harris [26], and van den Berg and Kesten [34]. For definitions of in-creasing events and disjoint occurrence, and for proofs of both inequalities, see [12].
Lemma 2.11. (Harris’s inequality) If A and B are increasing events then P p ( A ∩ B ) (cid:62) P p ( A ) · P p ( B ) . We write
A ◦ B for the event that A and B occur disjointly. Lemma 2.12. (The van den Berg–Kesten inequality) If A and B are increasingevents then P p ( A ◦ B ) (cid:54) P p ( A ) · P p ( B ) . We shall apply Harris’s inequality frequently throughout the paper, but the vanden Berg–Kesten inequality only once, in Lemma 8.9.3.
Voracity and quasi-stability
In Section 2 we mentioned that there were two important deterministic conceptsthat we needed in order to make our upper bound proofs work. These were thenotions of ‘voracious sets’ and ‘quasi-stable directions’. In this section we intro-duce and develop these ideas, in preparation for the proofs of the upper bounds ofTheorem 1.4 in the two sections to follow.
NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 17
Voracity and the infection of new lines.
We begin by studying sets ofinfected sites that are sufficient for stable half-planes to grow.
Definition 3.1.
Let u ∈ Q , and let Z ⊂ Z be a set of size | Z | = α ( u ). If[ H u ∪ Z ] ∩ (cid:96) u is infinite, then we say that Z is voracious for u .The definition of α ( u ) implies there exists at least one voracious set for every u ∈ S . We would like to show (see Lemma 3.4, below) that a bounded number ofvoracious sets on the u -side of a (finite) droplet D are sufficient to infect all buta bounded number of sites on the line adjacent to the u -side of D . The followingdefinition will be useful. Definition 3.2. A homothetic copy of a set S is a set Y = a + kS = (cid:8) y ∈ Z : y = a + kb for some b ∈ S (cid:9) for some a ∈ Z and non-zero k ∈ Z .Note that if a ∈ (cid:96) u , then a homothetic copy of (cid:96) + u is an infinite subset of the line (cid:96) u . As a warm-up for the (slightly technical) finite setting, let’s begin by provingthe infinite version of the statement we require. Lemma 3.3.
Let u ∈ Q be such that α ( u ) (cid:54) α and let Z be voracious for u . Then [ H u ∪ Z ] ∩ (cid:96) u contains a homothetic copy of (cid:96) + u .Proof. We may assume that u is stable, since otherwise the lemma is trivial, andthat [ H u ∪ Z ] contains infinitely many sites of (cid:96) + u , since Z is voracious. Since Z isfinite, there exists a ∈ Z such that [ H u ∪ Z ] ⊂ H u ( a ). Recall from (4) the definitionof ν , and partition H u ( a ) \ H u into disjoint congruent rectangles . . . , R − , R , R , . . . ,each of the same width s > ν , with R i immediately to the right of R i − for each i ∈ Z , and such that Z ⊂ R , noting that this is possible if s is sufficiently large.Set L i = R i ∩ [ H u ∪ Z ] for each i ∈ Z .Now, the condition that [ H u ∪ Z ] ∩ (cid:96) + u is infinite and the definition of ν togetherimply that L i is non-empty for every i (cid:62)
0. Since there are only finitely manypossible configurations for L i , there exist j (cid:62) r (cid:62) L j = L j + r .Furthermore, since the set L i is the final configuration inside R i , and since if i (cid:62) no sites to the right of R i (outside of H u ) are initially infected, it must be thatif i (cid:62) L i +1 depends only on L i . It follows that (cid:0) L j , . . . , L j + r − (cid:1) = (cid:0) L j + r , . . . , L j +2 r − (cid:1) = (cid:0) L j +2 r , . . . , L j +3 r − (cid:1) = . . . , and this is sufficient to prove the lemma. (cid:3) By taking suitable translates of the voracious set Z in Lemma 3.3, it is clear thatwe can infect the whole of either (cid:96) + u or (cid:96) − u . We can then use Lemma 2.8 to returnback along the line and infect the rest of (cid:96) u .Let us now turn to the finite setting, applicable to the growth of a new row onthe side of a droplet. Recall that we denote by ∂ ( D, u ) the u -side of a T -droplet D . In the lemma below we will also need the following notion: define the u -outside ofa T -droplet D to be the set ∂ ◦ ( D, u ) of points of (cid:96) u ( i + 1) that lie within distance1 of the convex hull of D , where ∂ ( D, u ) = D ∩ (cid:96) u ( i ). Let us also say that a set Z lies above the u -side of D if its orthogonal projection onto the continuous line(perpendicular to u ) through ∂ ( D, u ) intersects the convex hull of ∂ ( D, u ).The following lemma says that a bounded number of voracious sets are sufficient(together with H u ) to infect all but a bounded number of sites of the u -outside ofa droplet D , and that moreover we may choose any suitable translation of eachvoracious set. Lemma 3.4.
Let
T ⊂ Q be a finite set, let u ∈ T satisfy α ( u ) (cid:54) α , and let Z be voracious for u . Then there exist µ > , r ∈ N , and b ∈ (cid:96) u , such that for every T -droplet D , there exist a , . . . , a r ∈ ∂ ◦ ( D, u ) such that the following holds.Suppose that k , . . . , k r ∈ Z are such that Z + a j + k j b is above the u -side of D ,and at distance at least µ from the corners of D , for every (cid:54) j (cid:54) r . Then the set (cid:2) D ∪ ( Z + a + k b ) ∪ · · · ∪ ( Z + a r + k r b ) (cid:3) contains all elements of ∂ ◦ ( D, u ) at distance at least µ from the corners.Proof. First, choose a constant µ > X ∈ U and u ∈ T , we have y + ( X ∩ H u ) ⊂ D for every T -droplet D , andevery y ∈ ∂ ◦ ( D, u ) at distance at least µ from the corners. Now fix u ∈ T with α ( u ) (cid:54) α , and let Z be voracious for u , so (without loss of generality) we mayassume that [ H u ∪ Z ] contains infinitely many sites on the line (cid:96) + u . Define thesequence . . . , R − , R , R , . . . of rectangles (each of constant width s > ν ) as in theproof of Lemma 3.3, and set L i = R i ∩ [ H u ∪ Z ]for each i ∈ Z . Recall that Z ⊂ R , and define t = min (cid:8) t (cid:62) L ⊂ ( H u ∪ Z ) t (cid:9) ,i.e., the number of steps of the U -bootstrap process it takes to infect L , startingfrom H u ∪ Z . Since ‘information’ can only travel distance ν in one step of theprocess, it follows that if R is at distance at least t ν + µ from the corners of D ,then L ⊂ [ D ∪ Z ]. Next, for each i (cid:62)
1, define t i = min (cid:8) t (cid:62) L i ⊂ ( H u ∪ L i − ) t (cid:9) i.e., the number of steps of the U -bootstrap process it takes to infect L i , startingfrom H u ∪ L i − . Note that t i is finite, and moreover, since the L j are periodic thereexists a constant T such that t i (cid:54) T for every i (cid:62)
1. Therefore, if L i − ⊂ [ D ∪ Z ]and R i is at distance at least T ν + µ from the corners of D , then L i ⊂ [ D ∪ Z ].It follows that [ D ∪ Z ] ∩ ∂ ◦ ( D, u ) contains the intersection with ∂ ◦ ( D, u ) of a homo-thetic copy of (cid:96) + u with bounded difference. More precisely, there exists a ∈ ∂ ◦ ( D, u )and b ∈ (cid:96) u , where (cid:107) a − Z (cid:107) and (cid:107) b (cid:107) are both at most some constant depending on u and Z , but not on D , such that [ D ∪ Z ] ∩ ∂ ◦ ( D, u ) contains every element of a + b(cid:96) + u that is in ∂ ◦ ( D, u ), and at distance at least
T ν + µ from the corners of D . NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 19
Hence there exist r ∈ N (depending on b ) and a , . . . , a r ∈ ∂ ◦ ( D, u ) such that thefollowing holds: if k , . . . , k r ∈ Z are such that the set Z + a j + k j b is above the u -side of D and sufficiently far from the corners of D for every 1 (cid:54) j (cid:54) r , then theset Y := (cid:2) D ∪ ( Z + a + k b ) ∪ · · · ∪ ( Z + a r + k r b ) (cid:3) contains ν consecutive elements of ∂ ◦ ( D, u ) at distance at least µ from the corners.Now, by Lemma 2.8 there exist update rules X + and X − contained in H u ∪ (cid:96) + u and H u ∪ (cid:96) − u respectively. Note that X + \ H u is contained in the first ν sites of (cid:96) + u , andsimilarly for X − \ H u and (cid:96) − u . Hence Y in fact contains all elements of ∂ ◦ ( D, u ) thatare at distance at least µ from the corners, as required. (cid:3) As a consequence of Lemma 3.4, one would expect that a T -droplet D would‘grow by one step in direction u ’ with probability at least (cid:0) − (1 − p α ) Ω( m ) (cid:1) O (1) , where m is the length of the side of D corresponding to u . This is almost true;however, as the presence of the constant µ in Lemma 3.4 suggests, we have a problemnear the corners of D : we may need sites not in D but still below the (extended) u -side of D in order to infect the last O (1) sites. We resolve this problem usinganother idea from [13]: that of quasi-stable directions.3.2. Quasi-stability.
In many of the simpler bootstrap models, the droplets usedas bases for growth are taken with respect to the set of stable directions. Dropletsfor the 2-neighbour model are rectangles – or, put another way, they are takenwith respect to the set S = { e , − e , e , − e } of stable directions. Similarly, forbalanced threshold models with symmetric star-neighbourhoods droplets can betaken with respect to the set of stable directions, and the droplets are therefore2 k -gons consisting of pairs of parallel sides, for some k (cid:62)
2. In this case S -dropletsare suitable bases for growth because, when new infections spread in both directionsalong each edge of the droplet, the set that results is a new, slightly larger droplet.The same is not true in general: indeed, as noted above, we may fail to infectsome of the sites near the corners of D due to boundary effects. The solution to thisproblem as used by Bollob´as, Smith and Uzzell [13] was to introduce a number of quasi-stable directions , which are not stable directions, but which nevertheless aretreated as such. Thus, droplets are taken with respect to a certain superset of thestable set. For a comprehensive discussion of quasi-stability, we refer the reader toSection 5.1 of [13].The next lemma is Lemma 5.3 of [13]. Since the lemma is so fundamental to theproofs of the upper bounds of Theorem 1.4, we give the (short) proof here (whichis also similar to that of Lemma 2.7) in full. Recall that [ u, v ] denotes the interval We say that Y is a symmetric star-neighbourhood if x ∈ Y implies that that − x ∈ Y , andmoreover that every vertex of Z on the straight line between x and − x is in Y . of directions in S between u to v , taken anticlockwise starting from u . Given aset T ⊂ S , we say that u and v are consecutive elements of T if u (cid:54) = v and T ∩ [ u, v ] = { u, v } (note that the order of u and v matters in this definition). Lemma 3.5.
There exists a finite set
Q ⊂ Q such that for every pair u, v ofconsecutive elements of S ∪ Q there exists an update rule X such that X ⊂ (cid:0) H u ∪ (cid:96) u (cid:1) ∩ (cid:0) H v ∪ (cid:96) v (cid:1) . Proof.
Form Q by taking the two unit vectors u and − u perpendicular to x (con-sidered as a vector) for every site x ∈ X and every update rule X ∈ U . Formally, Q := (cid:91) X ∈U (cid:91) x ∈ X (cid:8) u ∈ S : (cid:104) u, x (cid:105) = 0 (cid:9) .(cid:96) v (cid:96) u v u(cid:96) w w xw (cid:48) (cid:0) H u ∪ (cid:96) u (cid:1) ∩ (cid:0) H v ∪ (cid:96) v (cid:1) Figure 2.
Since w is unstable, there exists X ∈ U with X ⊂ H w . If x ∈ X lies in the region between (cid:96) w and (cid:96) u , then the direction w (cid:48) wouldbe in Q , by construction, which contradicts u and v being consecutivein S ∪ Q . Thus X ⊂ ( H u ∪ (cid:96) u (cid:1) ∩ (cid:0) H v ∪ (cid:96) v (cid:1) , as required. Note thatthe figure is completely general: [ u, v ] is at most a semicircle, by thedefinition of S Q .Now suppose u and v are consecutive elements of S ∪ Q and let w ∈ [ u, v ] \ { u, v } .Since w is not stable, there exists an update rule X ⊂ H w . Suppose the conclusionof the lemma fails, so X (cid:54)⊂ (cid:0) H u ∪ (cid:96) u (cid:1) ∩ (cid:0) H v ∪ (cid:96) v (cid:1) . Then without loss of generality there exists x ∈ X such that (cid:104) x, v (cid:105) < (cid:104) x, u (cid:105) >
0. But this implies that there exists w (cid:48) ∈ S perpendicular to x with w (cid:48) ∈ [ u, v ] \{ u, v } , contradicting the construction of Q . (See Figure 2.) (cid:3) It follows immediately from the lemma that when droplets are taken with respectto suitable finite subsets of
S ∪ Q , there are rules that allow the droplets to growalong their sides all the way to the corners: droplets grow into droplets . NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 21 The upper bound for balanced families
In this section we shall prove the following theorem, which is the upper bound ofTheorem 1.4 for balanced families.
Theorem 4.1.
Let U be critical and balanced. Then p c ( Z n , U ) = O (cid:18) n (cid:19) /α . Recall that if U is balanced then there exists a closed semicircle C ⊂ S such that α ( u ) (cid:54) α for all u ∈ C . Since α ( u ) < ∞ for every u ∈ C , every stable direction u ∈ C must be isolated, by Lemma 2.6. This implies the existence of a closed arc C (cid:48) such that C (cid:40) C (cid:48) (cid:40) S and such that α ( u ) (cid:54) α for all u ∈ C (cid:48) . We write u + forthe left endpoint of C (cid:48) and u − for the right endpoint; we may assume that these arerational.Let Q be the set of quasi-stable directions given by Lemma 3.5 and set S Q := (cid:0) S ∪ Q ∪ { u − , u + } (cid:1) ∩ C (cid:48) and S (cid:48) Q := S Q \ { u − , u + } . These sets are finite, since Q and S ∩ C (cid:48) are both finite by construction. Throughoutthis section droplets will be taken with respect to the set S Q .Choose a collection of vectors { a u ∈ R : u ∈ S Q } and sufficiently large positiveconstants { d u > u ∈ S (cid:48) Q } such that the sequence of S Q -droplets D m := (cid:92) u ∈{ u − ,u + } H u ( a u ) ∩ (cid:92) u ∈S (cid:48) Q H u ( a u + md u u ) (6)for m = 0 , , , . . . have the following properties (see Figure 3):(i) D is sufficiently large relative to the d u ;(ii) for every m (cid:62) u, v ∈ S (cid:48) Q , the intersection ofthe lines (cid:96) u + a u + md u u and (cid:96) v + a v + md v v lies on a (continuous) line L + u = L − v ;(iii) the lines L + u all intersect at the point x ∈ R , which is also the intersectionpoint of the sides of D corresponding to u − and u + .The key lemma in our proof of Theorem 4.1 will be the following bound on theprobability that a droplet grows by a constant distance. Lemma 4.2.
Let m ∈ N . Then P p (cid:16) D m ⊂ (cid:2) D m − ∪ ( D m +1 ∩ A ) (cid:3)(cid:17) (cid:62) (cid:0) − (1 − p α ) Ω( m ) (cid:1) O (1) . Note that the constants implicit in the right-hand side of the inequality abovedepend on our choice of droplets, and hence on U , but not on the probability p .Before proving Lemma 4.2, let us show that it implies the following lower bound onthe probability that a large droplet is almost internally filled. These are discrete lines and may have empty intersection. If this is the case then we meaninstead the intersection of the corresponding continuous lines; this may not be an element of Z . D D \ D x L + u = L − v u + u − vu Figure 3.
The sequence of droplets D ⊂ D ⊂ D ⊂ · · · . Lemma 4.3.
Let m ∈ N . Then P p (cid:16) D m ⊂ (cid:2) D m +1 ∩ A (cid:3)(cid:17) (cid:62) exp (cid:16) − O (cid:0) p − α (cid:1)(cid:17) . Proof.
Noting that all the events we are considering are increasing, it follows fromHarris’s inequality (Lemma 2.11) that P p (cid:16) D m ⊂ (cid:2) D m +1 ∩ A (cid:3)(cid:17) (cid:62) P p (cid:18) I ( D ) ∩ m (cid:92) k =1 (cid:110) D k ⊂ (cid:2) D k − ∪ ( D k +1 ∩ A ) (cid:3)(cid:111)(cid:19) (cid:62) P p (cid:0) I ( D ) (cid:1) m (cid:89) k =1 P p (cid:16) D k ⊂ (cid:2) D k − ∪ ( D k +1 ∩ A ) (cid:3)(cid:17) . Thus, by Lemma 4.2, we have P p (cid:16) D m ⊂ (cid:2) D m +1 ∩ A (cid:3)(cid:17) (cid:62) p O (1) ∞ (cid:89) k =1 (cid:0) − (1 − p α ) Ω( k ) (cid:1) O (1) (cid:62) p O (1) exp (cid:18) − O (1) ∞ (cid:88) k =1 − log (cid:16) − e − Ω( p α k ) (cid:17)(cid:19) (cid:62) p O (1) exp (cid:18) − O (cid:0) p − α (cid:1) (cid:90) ∞ − log (cid:0) − e − z (cid:1) dz (cid:19) (cid:62) exp (cid:16) − O (cid:0) p − α (cid:1)(cid:17) , where for the final inequality we used the fact that (cid:82) ∞ − log (cid:0) − e − z (cid:1) dz < ∞ . (cid:3) From here, the deduction of Theorem 4.1 is straightforward.
Proof of Theorem 4.1.
Let λ be a sufficiently large constant, and set p = (cid:18) λ log n (cid:19) /α . NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 23
As usual, A is a p -random subset of Z n . We shall show that [ A ] = Z n with highprobability as n → ∞ , which is more than enough to prove the theorem.To avoid some technical issues, let us ‘sprinkle’ the initially infected sites in tworounds; that is, we take A (1) and A (2) to be independent p -random subsets of Z n ,and redefine the set of initially infected sites to be A = A (1) ∪ A (2) . This means weare actually including sites in A with probability 2 p − p , but this does not matterbecause we have freedom over the infection probability up to a constant factor. Weuse the first round of sprinkling to find an almost internally filled copy of D m , where m := (log n ) , and the second round to show that the copy of D m grows (with highprobability) to fill the torus.Let us define the following events: E := (cid:91) x ∈ Z n (cid:110) x + D m ⊂ (cid:2) ( x + D m +1 ) ∩ A (1) (cid:3)(cid:111) is the event that x + D m is ‘almost internally filled’ by A (1) , for some x ∈ Z , and F ( x ) := (cid:110) Z n ⊂ (cid:2) ( x + D m ) ∪ A (2) (cid:3)(cid:111) is the event that Z n is internally filled by ( x + D m ) ∪ A (2) . The events E and F ( x ) areindependent, so by the comments above, it will suffice to show that P p ( E c ) = o (1)and that P p (cid:0) F ( x ) c (cid:1) = o (1) for each fixed x ∈ Z n .To bound P p ( E c ), observe first that there exists a collection of Ω (cid:0) n /m (cid:1) sites x ∈ Z n such that the sets x + D m +1 are pairwise disjoint. By Lemma 4.3, it followsthat P p (cid:0) E c (cid:1) (cid:54) (cid:16) − exp (cid:0) − O ( p − α ) (cid:1)(cid:17) Ω( n /m ) (cid:54) exp (cid:16) − n o (1) e − O (log n ) /λ (cid:17) = o (1)since λ is sufficiently large.To bound P p (cid:0) F ( x ) c (cid:1) , observe that x + D λn = Z n if λ is sufficiently large. ByLemma 4.2 and Harris’s inequality (which we need because the ‘wrap-around’ effectof the torus causes loss of independence), it follows that, for any x ∈ Z n , P p (cid:0) F ( x ) (cid:1) (cid:62) λn +1 (cid:89) k = m (cid:16) − (cid:0) − p α (cid:1) Ω( k ) (cid:17) O (1) (cid:62) (cid:16) − (cid:0) − p α (cid:1) Ω( m ) (cid:17) O ( n ) (cid:62) exp (cid:16) − O (cid:0) e − Ω( p α m ) · n (cid:1)(cid:17) = 1 − o (1) , as claimed. By the comments above, this completes the proof of the theorem. (cid:3) Our only remaining task is to prove Lemma 4.2. Having already established thedeterministic lemmas of the previous section, the idea of the proof is simple. Inorder to grow from D m to D m +1 it is sufficient for a bounded number of events tooccur, each event having failure probability at most (1 − p α ) Ω( m ) . These events areall very loosely speaking of the form ‘there exists in A a translate of a given set of α sites somewhere along one of the edges of the droplet’. Since any set of α sites is a subset of A with probability p α , and since there are Ω( m ) possible disjoint translatesof that set, we obtain the desired bound on the probability.For the sake of completeness, we now present a rigorous proof of Lemma 4.2. Webegin by giving a name to the sets that we shall use to grow the droplets. Let D bean S Q -droplet with D m − ⊂ D ⊂ D m , let u ∈ S (cid:48) Q , and let Z be an arbitrary (butfixed) set that is voracious for u . Let µ > r ∈ N , b ∈ (cid:96) u and a , . . . , a r ∈ ∂ ◦ ( D, u )be given by Lemma 3.4, applied to S Q , u , Z and D (so µ , r and b depend on u and Z , but not D ). We will use the following sets in order to infect ∂ ◦ ( D, u ). Definition 4.4.
For each 1 (cid:54) j (cid:54) r , an ( α, u, j ) -cluster for D is a set of the form Z + a j + kb for some k ∈ Z . We say that an ( α, u, j )-cluster for D is suitable if it lies above the u -side of D , and at distance at least µ from the corners of D .The following deterministic lemma is a straightforward consequence of Lemma 3.4and 3.5. Lemma 4.5.
For each (cid:54) j (cid:54) r , let Z ( j ) be a suitable ( α, u, j ) -cluster for D . Then ∂ ◦ ( D, u ) ⊂ (cid:104) D ∪ r (cid:91) j =1 Z ( j ) (cid:105) (7) Proof.
By Lemma 3.4, the right-hand side of (7) contains all sites in ∂ ◦ ( D, u ) atdistance at least µ from the corners. In particular, since D was assumed to besufficiently large, it follows that an interval of at least ν consecutive sites of ∂ ◦ ( D, u )are infected. Now, using Lemma 3.5, we can also infect the elements of ∂ ◦ ( D, u )near the corners of D . (cid:3) Lemma 4.2 is a simple consequence of Lemma 4.5.
Proof of Lemma 4.2.
Observe that for every S Q -droplet D with D m − ⊂ D ⊂ D m ,and every u ∈ S (cid:48) Q and 1 (cid:54) j (cid:54) r = r ( u ), there are at least Ω( m ) disjoint suitable( α, j )-clusters. Since each ( α, j )-cluster is contained in A with probability p α , itfollows by Lemma 4.5 and Harris’s lemma that P p (cid:16) ∂ ◦ ( D, u ) ⊂ (cid:2) D ∪ ( D m +1 ∩ A ) (cid:3)(cid:17) (cid:62) (cid:0) − (1 − p α ) Ω( m ) (cid:1) O (1) . Hence, since a bounded number of steps of this form suffice to infect D m , usingHarris’s lemma once again we obtain P p (cid:16) D m ⊂ (cid:2) D m − ∪ ( D m +1 ∩ A ) (cid:3)(cid:17) (cid:62) (cid:0) − (1 − p α ) Ω( m ) (cid:1) O (1) , as required. (cid:3) NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 25 The upper bound for unbalanced families
In this final section on upper bounds we prove the following general theorem,which in particular implies the upper bound in Theorem 1.4 for unbalanced families.
Theorem 5.1.
Let U be a critical update family. Then p c ( Z n , U ) = O (cid:18) (log log n ) log n (cid:19) /α . The theorem does not require the hypothesis that U is unbalanced, although ofcourse it is only under this assumption that the result is tight up to the implicitconstant. It may be helpful in this section to think of U as being unbalanced, eventhough this is not strictly necessary.By the definition of α , there exists an open semicircle C ⊂ S such that α ( u ) (cid:54) α for all u ∈ C . Let u ⊥ be the midpoint of C , let u ∗ and − u ∗ be the left and rightendpoints of C respectively, and note that α + ( u ∗ ) < ∞ (and similarly α − ( − u ∗ ) < ∞ ) by Lemma 2.7. Thus, by Lemma 2.8, there exists a finite set of consecutive sites Z ⊂ (cid:96) u ∗ such that (cid:96) + u ∗ ⊂ [ H u ∗ ∪ Z ]. Define α ∗ to be the (minimum, say) size of sucha set Z .Let Q be the set of quasi-stable directions given by Lemma 3.5, and set S Q := (cid:0) ( S ∪ Q ) ∩ C (cid:1) ∪ { u ∗ , − u ∗ , − u ⊥ } and S (cid:48) Q := ( S ∪ Q ) ∩ C. As in the previous section, both of these sets are finite. In this section all dropletswill be S Q -droplets. Since the growth process will predominantly take place indirections parallel to the vectors u ⊥ and u ∗ , to simplify the notation we rotate thelattice Z so that u ∗ is directed vertically upwards. The discrete rectangle withopposite corners ( a, b ) and ( c, d ) is thus defined to be R (cid:0) ( a, b ) , ( c, d ) (cid:1) := (cid:8) xu ⊥ + yu ∗ ∈ Z : x, y ∈ R , a (cid:54) x (cid:54) c and b (cid:54) y (cid:54) d (cid:9) . The sequences of droplets will be defined in terms of the following quantities: m ( p ) := λ p α log 1 p , m ( p ) := p − λ , m ( p ) = p α ∗ m ( p ) and m ( p ) := λ n, where λ (cid:29) λ (cid:29) n = n ( p ), to bespecified later (see (9)), satisfies log n (cid:54) p − λ / . (8)Let R := R (cid:16) (0 , , (cid:0) λ , m ( p ) (cid:1)(cid:17) , R := R (cid:16) (0 , , (cid:0) m ( p ) + λ , m ( p ) (cid:1)(cid:17) ,R := R (cid:16)(cid:0) m ( p ) , (cid:1) , (cid:0) m ( p ) + λ , m ( p ) + m ( p ) (cid:1)(cid:17) , and R := R (cid:16)(cid:0) m ( p ) , (cid:1) , (cid:0) m ( p ) , m ( p ) + m ( p ) (cid:1)(cid:17) . be rectangles, and let T := (cid:110) xu ⊥ + yu ∗ ∈ Z : 0 (cid:54) x (cid:54) m ( p ) + λ and 0 (cid:54) y − m ( p ) (cid:54) p α ∗ x (cid:111) be a triangle; see Figure 4. 1 2 3 u ∗ u ⊥ R R R R Tλ m ( p ) m ( p ) m ( p ) D (1)0 D (3)0 Figure 4.
The growth mechanism in the unbalanced setting.For technical reasons, we also need to use the rectangles R (cid:48) := R (cid:16) (0 , , (cid:0) m ( p ) , m ( p ) (cid:1)(cid:17) and R (cid:48) := R (cid:16)(cid:0) m ( p ) , (cid:1) , (cid:0) m ( p ) , m ( p ) + m ( p ) (cid:1)(cid:17) . which are roughly twice as long as R and R respectively.Figure 4 illustrates the growth mechanism we use to prove Theorem 5.1. It comesin five stages, and, as in the previous section, we use sprinkling to maintain inde-pendence between the different stages. • Stage 0.
We find a copy of R contained in A . • Stage 1.
The infection spreads in the direction u ⊥ from R and fills the rec-tangle R . This occurs in a similar way to growth in balanced models, exceptthat the rows are not increasing in size. • Stage 2.
The infection spreads in the direction u ∗ from R , using infected sitesin the triangle T to fill R . • Stage 3.
Exactly as in Stage 1, the infection spreads in the direction u ⊥ from R to fill R . • Stage 4.
Now, R is a ‘strip’ that wraps around the torus and either covers Z n , or returns to its starting point. In the latter case, the infection spreads indirection u ∗ from R (like in Stage 2) to infect the rest of Z n . NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 27
Our task is to make the above sketch precise. We postpone the proof of thefollowing key lemma until later in the section.
Lemma 5.2.
The event (cid:110) R ⊂ (cid:2) R ∪ (cid:0) ( R (cid:48) ∪ T ∪ R (cid:48) ) ∩ A (cid:1)(cid:3)(cid:111) occurs with high probability as p → . From here, the deduction of Theorem 5.1 is relatively straightforward, the maincomplication being the possibility that R (cid:54) = Z n . Proof of Theorem 5.1.
The proof is similar to the proof of Theorem 4.1 for balancedfamilies. Let λ > p = (cid:18) λ (log log n ) log n (cid:19) /α . (9)We shall show that [ A ] = Z n with high probability as n → ∞ .As before, we sprinkle in two rounds, each round using probability p (which, alsoas before, is permissible, if a slight abuse of notation), and denote by A (1) and A (2) the sites infected in each round. There are (crudely) at least n choices of x ∈ Z n such that the sets x + R are disjoint, and the probability that x + R (cid:54)⊂ A (1) for allsuch x is at most (cid:0) − p O ( m ( p )) (cid:1) n . (10)Noting that p O ( m ( p )) = exp (cid:32) − O (1) p α (cid:18) log 1 p (cid:19) (cid:33) (cid:62) √ n since λ is sufficiently large, it follows that (10) tends to 0 as n → ∞ .Now fix x such that x + R ⊂ A (1) , and in fact without loss of generality let usassume that x = . By Lemma 5.2, P p (cid:16) R ⊂ (cid:2) R ∪ (cid:0) ( R (cid:48) ∪ T ∪ R (cid:48) ) ∩ A (2) (cid:1)(cid:3)(cid:17) = 1 − o (1) , since the condition on n in (8) holds with our definition of p . If R = Z n thenwe are done; otherwise, it follows (since λ is sufficiently large) that R is a ‘strip’that wraps around Z n a positive integer number of times before returning to itsstarting point. Thus, in order to infect the remaining sites in Z n , it is enough, bythe definition of α ∗ , for the following event to occur: every line in Z n parallel to u ⊥ contains α ∗ consecutive sites. Indeed, this would ensure that the remaining linesabove the strip R are infected one-by-one. Since each line has length at least Ω( n ),and there are at most O ( n ) lines, the probability that this event fails is at most O ( n ) · (cid:0) − p α ∗ (cid:1) Ω( n ) = o (1) , and this completes the proof of the theorem. (cid:3) We have reduced our task to that of proving Lemma 5.2. As in the previoussection, given the framework of voracity and quasi-stability from Section 3, the ideaof the proof is simple. In Stage 1 of the process, the probability of advancing aconstant number of steps is (cid:0) − (1 − p α ) Ω( m ( p )) (cid:1) O (1) (cid:54) (1 − p Ω( λ ) ) O (1) . Since λ (cid:29) λ , the set should grow to fill R , and for similar reasons, the infectionspreads out rightwards from R to fill R . Both of these steps are almost the sameas the corresponding part of the proof for balanced models. The growth upwardsfrom R through T to fill R is a little different. Since the infection might onlyspread rightwards when advancing row-by-row in the u ∗ direction, each set of α ∗ consecutive infected sites we find when growing upwards through T from R mustlie to the right of the previous set. Nevertheless, the probability of filling T (exceptpossibly for a small number of sites near the diagonal) is at least (cid:0) − (1 − p α ∗ ) Ω( p − α ∗ ) (cid:1) O ( m ( p )) = 1 − o (1) . Since the proof is easy but notationally a little technical, we encourage the readerwho is satisfied with the sketch above to skip ahead to Section 6.We define two sequences of droplets as in (6), except with u + = u ∗ and u − = − u ∗ (so the corresponding lines are now parallel), and with − u ⊥ added to the set ofquasi-stable directions. Specifically, for each m ∈ Z and each i ∈ { , } , define the S Q -droplets D ( i ) m := R i ∩ (cid:92) u ∈S (cid:48) Q H u (cid:0) a u + md u u (cid:1) (11)for some { a u ∈ R : u ∈ S (cid:48) Q } and sufficiently large positive constants { d u > u ∈S (cid:48) Q } such that: • R i − ⊂ D ( i )0 ; • for every consecutive pair u, v ∈ S (cid:48) Q , there exists a horizontal line L + u = L − v (that is, one parallel to u ⊥ ) that intersects R i , such that for every m ∈ Z , theintersection of (cid:96) u + a u + md u u and (cid:96) v + a v + md v v lies on L + u = L − v ; • for each u ∈ S (cid:48) Q and each m ∈ N , the u -side of D ( i ) m has size Ω (cid:0) m i ( p ) (cid:1) .Note that we shall also need to use D ( i ) m for those negative values of m for which thedroplet is non-empty, as well as for positive values of m .The following lemma is essentially Lemma 4.2 applied to the droplets D ( i ) m , andso the proof is omitted. Lemma 5.3.
Let i ∈ { , } and m ∈ Z . Then P p (cid:16) D ( i ) m ⊂ (cid:2) R i − ∪ D ( i ) m − ∪ ( D ( i ) m +1 ∩ A ) (cid:3)(cid:17) (cid:62) (cid:0) − (1 − p α ) Ω( m i ( p )) (cid:1) O (1) . (cid:3) We now complete the proof of Lemma 5.2.
NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 29
Proof of Lemma 5.2.
We shall show that the event (cid:110) R ⊂ (cid:2) R ∪ ( R (cid:48) ∩ A ) (cid:3)(cid:111) ∩ (cid:110) R ⊂ (cid:2) R ∪ ( T ∩ A ) (cid:3)(cid:111) ∩ (cid:110) R ⊂ (cid:2) R ∪ ( R (cid:48) ∩ A ) (cid:3)(cid:111) occurs with high probability as p →
0, which clearly implies the lemma. We begin bydeducing from Lemma 5.3 that the first and third parts of this event occur with highprobability as p →
0. To see this, let i ∈ { , } and observe that there exists m ∈ N such that R i ⊂ D ( i ) m ⊂ D ( i ) m +1 ⊂ R (cid:48) i , if p is sufficiently small, where m = O (cid:0) m i +1 ( p ) (cid:1) .It follows from Lemma 5.3 that P p (cid:16) D ( i ) m ⊂ (cid:2) R i − ∪ ( D ( i ) m +1 ∩ A ) (cid:3)(cid:17) (cid:62) (cid:16) − (1 − p α ) Ω( m i ( p )) (cid:17) O ( m i +1 ( p )) (cid:62) exp (cid:16) − O (cid:0) m i +1 ( p ) (cid:1) · exp (cid:0) − Ω( m i ( p ) · p α ) (cid:1)(cid:17) = 1 − o (1)as p →
0. Indeed, we have exp (cid:0) − Ω( m ( p ) · p α ) (cid:1) = p Ω( λ ) = o (cid:0) /m ( p ) (cid:1) , andexp (cid:0) − Ω( m ( p ) · p α ) (cid:1) = exp (cid:0) − Ω( p − λ +2 α ∗ + α ) (cid:1) < n = o (cid:18) m ( p ) (cid:19) , where we used our assumptions that λ (cid:29) λ (cid:29) n (cid:54) p − λ / .It remains to show that the event (cid:110) R ⊂ (cid:2) R ∪ ( T ∩ A ) (cid:3)(cid:111) occurs with high probability as p →
0. To do so, consider the set U i of the leftmost p − α ∗ sites of T ∩ (cid:96) u ∗ ( i ) for each line (cid:96) u ∗ ( i ) that intersects T . Now, suppose that, forevery such line, the middle p − α ∗ / U i contain a set of α ∗ consecutive sitesof A . Then R ⊂ (cid:2) R ∪ ( T ∩ A ) (cid:3) , by the definition of α ∗ . But this has probabilityat least (cid:0) − (1 − p α ∗ ) p − α ∗ / α ∗ (cid:1) m ( p ) (cid:62) exp (cid:16) − p − λ exp (cid:0) − p − α ∗ / ) (cid:1)(cid:17) = 1 − o (1) , as required. (cid:3) Approximately internally filled sets
In this section we lay the groundwork for the proofs of the lower bounds of The-orem 1.4 by defining and proving basic properties of three of our key tools: the covering , spanning and iceberg algorithms. These should all be thought of as waysof using droplets to approximate the closure of A under the U -bootstrap process.The covering algorithm, which we introduce in Section 6.1, replaces the rectanglesprocess in the balanced case, and allows us to find in Z n (if A percolates) a dropletof size about log n containing Ω(log n ) disjoint, strongly connected subsets of A of size α . For unbalanced models, we use the spanning algorithm, introduced inSection 6.2, to find an internally spanned critical droplet and to construct an iteratedsequence of ‘hierarchies’ for this droplet. For models with drift, we will in additionrequire the iceberg algorithm, which we will introduce in Section 6.3, in order to bound the range of the U -bootstrap process in certain directions with the help ofhalf-planes.Having completed the proofs of the upper bounds of Theorem 1.4, we no longerhave any need for quasi-stable directions. In fact, henceforth all droplets will beassumed to be taken with respect to one of two specific finite sets of stable directions,according to whether U is balanced or unbalanced. The existence of these sets isverified in the next two lemmas. For u ∈ Q , let¯ α ( u ) := min (cid:8) α + ( u ) , α − ( u ) (cid:9) , (12)so ¯ α ( u ) = α ( u ) if and only if α + ( u ) and α − ( u ) are either both finite or both infinite. Lemma 6.1. If U is a critical update family, then there exists a finite set S B ⊂ Q such that:(i) ¯ α ( u ) (cid:62) α for every u ∈ S B ; and(ii) S B ∩ C (cid:54) = ∅ for every open semicircle C ⊂ S . Although S B exists for any critical update family U , we will only need this familywhen U is balanced. We remark that condition ( ii ) is equivalent to the origin lyingin the interior of the convex hull of S B , and also to S B -droplets being finite. Proof of Lemma 6.1.
Observe first that, by Definition 1.2, there exists a finite set
T ⊂ Q , satisfying condition ( ii ), such that α ( u ) (cid:62) α for all u ∈ T . Now, recallfrom Lemma 2.7 that α ( u ) = ¯ α ( u ) unless u is an endpoint of a non-trivial intervalof S . However, if this is the case for some u ∈ T , then there exist vectors u (cid:48) ∈ Q with ¯ α ( u (cid:48) ) = ∞ arbitrarily close to u . Choosing such a u (cid:48) sufficiently close to u ,and replacing u by u (cid:48) , we see that condition ( ii ) still holds. Repeating this for each u ∈ T with α ( u ) (cid:54) = ¯ α ( u ), we obtain a set with the desired properties. (cid:3) It is easy to see that we may in fact take S B to have size 3, except when |S| = 4and the elements of S form the vertices of a parallelogram, in which case we maytake S B to have size 4. We shall not need this observation, however.We next show that a suitable collection of stable directions exists when U isunbalanced; the properties we need in this case are somewhat different. Lemma 6.2. If U is unbalanced then there exists a finite set S U ⊂ Q such that:(i) S U = { u ∗ , − u ∗ , u l , u r } for some u ∗ , u l , u r ∈ S such that u l lies in the opensemicircle to the left of u ∗ and u r in the open semicircle to the right;(ii) min (cid:8) α ( u ∗ ) , α ( − u ∗ ) (cid:9) (cid:62) α + 1 ; and(iii) min (cid:8) ¯ α ( u l ) , ¯ α ( u r ) (cid:9) (cid:62) α .Proof. Choose u ∗ satisfying condition ( ii ) using Lemma 2.9, and then u l and u r satisfying the remaining two conditions using Definition 1.2. In particular, notethat if one of the open semicircles bounded by u ∗ and − u ∗ contains an interval ofstable directions then we may choose any interior point of this interval, and if not,then α ( u ) = ¯ α ( u ) for every u in the open semicircle. (cid:3) NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 31
As mentioned before Lemma 6.1, we shall henceforth fix sets S B (if U is balanced)and S U (if U is unbalanced) with the above properties. We also make the followingdefinition, which we will use extensively. Definition 6.3.
Given an update family U and a finite set K ⊂ Z , we will write D ( K ) for the unique minimal S ∗ -droplet containing K , where S ∗ = S B if U isbalanced, and S ∗ = S U if U is unbalanced.Recall that in Section 2.5 we defined ν to be the diameter of U , ν = max (cid:110) (cid:107) x − y (cid:107) : x, y ∈ X ∪ { } , X ∈ U (cid:111) . We will need the following additional measure of the range of the process: ρ := sup (cid:110) (cid:107) y − Z (cid:107) : | Z | = α − , y ∈ (cid:2) H u ∪ Z (cid:3) \ H u , u ∈ S B (cid:111) , (13)where the supremum is taken over all choices of u , y and Z satisfying the statedconditions. In order to prove that ρ is finite, we will need the following extremallemma from [13]. Lemma 6.4 (Lemma 4.7 of [13]) . For any finite set Z ⊂ Z , the closure [ Z ] iscontained in a collection of disjoint S B -droplets, each of diameter O ( | Z | ) . To prove Lemma 6.4, simply place an S B -droplet on each element of Z , and thenrecursively unite any pair that lie within distance ν of each other by replacing themby the smallest S B -droplet that contains both (cf. Definition 6.6 and Lemma 6.9). Itis now not too difficult to deduce that ρ is finite whenever U is balanced; we recordthis important fact as the following lemma. Lemma 6.5. ρ < ∞ for every critical update family U .Proof. For each u ∈ S B , set ρ ( u ) = 0, and for i = 1 , . . . , α −
1, define ρ i ( u ) := sup (cid:110) (cid:107) y − Z (cid:107) : | Z | = i, y ∈ (cid:2) H u ∪ Z (cid:3) \ H u (cid:111) . We shall prove inductively that each ρ i ( u ) is finite. Indeed, let 1 (cid:54) i (cid:54) α −
1, andassume that ρ i − ( u ) is finite. Let a > H u ( au ) ∩ Z = ∅ . By Lemma 6.4, we have (cid:107) y − Z (cid:107) = O ( | Z | )for every y ∈ [ Z ] = (cid:2) H u ∪ Z (cid:3) \ H u , as required, where the last equality holds since a is sufficiently large. So let Z ⊂ Z with | Z | = i and H u ( au ) ∩ Z (cid:54) = ∅ . Now, if Z (cid:54)⊂ H u (cid:0) iau (cid:1) , then there exists a strip perpendicular to u of width 2 a that contains no element of Z , and separates some element of Z from H u . Since Z contains at least one elementof H u ( au ), it follows that there are elements of Z on both sides of this strip, and Note that this is obtained by taking a tangent line to K in each direction of S ∗ . these two sets of elements cannot interact, by the induction hypothesis (and since a is sufficiently large). It follows that we are also done in this case.We may therefore assume that Z ⊂ H u (cid:0) iau (cid:1) . Now, if [ H u ∪ Z ] \ H u is infinite thenit must contain an infinite number of elements of some line (cid:96) u ( j ) ⊂ H u (3 iau ) \ H u ,in which case there exists a translate Z (cid:48) of Z such that [ H u ∪ Z (cid:48) ] ∩ (cid:96) u is infinite.But this contradicts our assumption that i < α (cid:54) ¯ α ( u ) (since u ∈ S B , and usingLemma 6.1), so in fact [ H u ∪ Z ] \ H u is finite for every such Z .Finally, observe that if there is an element of Z at distance more than 2 ρ i − ( u ) + ν from all other elements of Z , then this element does not interact with the others (bythe induction hypothesis), in which case we are again done. But now there are onlya bounded number of choices for Z (up to translation by an element of (cid:96) u ), and foreach of these [ H u ∪ Z ] \ H u is finite, so it follows that ρ i ( u ) is finite, as required. (cid:3) The constant κ in Definition 2.3 of a strongly connected set will be defined dif-ferently according to whether U is balanced or unbalanced as follows: κ = κ ( U ) := (cid:40) ρ + ν if U is balanced, and3 ν if U is unbalanced. (14)Recall that sites x and y are said to be strongly connected if (cid:107) x − y (cid:107) (cid:54) κ .To simplify the presentation, we will work on the infinite lattice Z . However, itwill be clear that the algorithms and lemmas below can be easily modified to thesetting of the torus Z n , modulo some (easily resolved, but distracting ) technicalissues that arise when the droplets have diameter Θ( n ). Since the droplets in our ap-plications (see Sections 7 and 8.6) will all have diameter (log n ) O (1) , we can reassurethe concerned reader that these technical issues will not arise in practice.6.1. The covering algorithm: balanced families.
Throughout this subsectionwe assume that U is balanced, and that all droplets are taken with respect to S B . Wewill define the collection of α -covers of a finite set K , and use this definition to provetwo key lemmas: an ‘Aizenman–Lebowitz lemma’, which says that an α -covereddroplet contains α -covered droplets of all intermediate sizes, and an extremal lemma,which says that an α -covered droplet contains many disjoint ‘ α -clusters’. The proofsof both lemmas are straightforward applications of the covering algorithm.The key complication arising from the algorithm is that an α -cover of a set K doesnot necessarily contain the closure of K under the U -bootstrap process. However,an approximate version of this statement is true, and this is proved in Lemma 6.11.Roughly speaking, the lemma says that one can obtain (a superset of) the closure[ K ] from an α -cover of K via only ‘local’ modifications. For example, we could simply set D ( K ) = Z n for any set K of diameter larger than εn forsome small constant ε > NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 33
We define an α -cluster to be any strongly connected set of α sites. These will beour basic building blocks in the covering algorithm. Recall that if U is balanced,then D ( K ) denotes the unique minimal S B -droplet containing K . Definition 6.6 ( The α -covering algorithm ) . Let U be balanced. Suppose that weare given: • K , a finite set of infected sites in Z ; • B , . . . , B k , a maximal collection of disjoint α -clusters in K ; • D = { D , . . . , D k } , a collection of copies of a fixed, sufficiently large S B -droplet ˆ D , such that B j ⊂ D j for each j = 1 , . . . , k .Set t := 0 and repeat the following steps until STOP:1. If there are two droplets D ti , D tj ∈ D t and an x ∈ Z such that the set D ti ∪ D tj ∪ ( x + ˆ D ) (15)is strongly connected, then set D t +1 := (cid:0) D t \ { D ti , D tj } (cid:1) ∪ (cid:8) D ( D ti ∪ D tj ) (cid:9) , and set t := t + 1.2. Otherwise set T := t and STOP.The output of the algorithm is the family D := { D T , . . . , D Tk } , where k = k − T .Thus, at each step of the algorithm, we take two nearby droplets in our collection,and replace them by the smallest S B -droplet containing their union. Let us fix fromnow on a sufficiently large S B -droplet ˆ D as in the covering algorithm. In particular,in Lemma 6.11 we shall need that ˆ D contains a ball of radius 2 ακ . Definition 6.7.
We say that D = { D , . . . , D k } is an α -cover of a finite set K ⊂ Z if D is a possible output of the α -covering algorithm with input K . We say that adroplet D is α -covered by A if the single droplet D = { D } is an α -cover of D ∩ A .We will show (see Lemmas 6.9 and 6.12) that if [ A ] = Z n then there exists an α -covered droplet of diameter roughly log n , and that such a droplet must containat least Ω(log n ) disjoint α -clusters. It will then be relatively straightforward todeduce the lower bound in Theorem 1.4 for balanced update families, which we doin Section 7.The first important property of the α -covering algorithm is given by the followinglemma. We call this result an ‘Aizenman–Lebowitz lemma for α -covered droplets’,since the corresponding result for the 2-neighbour process was first proved in [1].Let λ be a sufficiently large constant, depending on ˆ D . Lemma 6.8.
Let D be an α -covered droplet. Then for every λ (cid:54) k (cid:54) diam( D ) there exists an α -covered droplet D (cid:48) ⊂ D such that k (cid:54) diam( D (cid:48) ) (cid:54) k . Proof.
The lemma is an immediate consequence of two simple observations: thatthe droplets D ti ∈ D t obtained during the α -covering algorithm are all α -covered,and that at each step of the algorithm,max (cid:8) diam( D ti ) : D ti ∈ D t (cid:9) at most triples in size, provided that this maximum is at least an absolute constant(depending on ˆ D ).To prove the first observation, simply run the algorithm on D ti ∩ A , using the same α -clusters and combining them in the same order. To prove the second, observe thatif droplets D ti and D tj are united in step t of the algorithm, then by definition thereexists x ∈ Z such that the distance between D ti and x + ˆ D , and that between D tj and x + ˆ D , are at most κ . Since for any two intersecting droplets D and D wehave the easy geometric inequalitydiam (cid:0) D ( D ∪ D ) (cid:1) (cid:54) diam( D ) + diam( D ) , it follows that diam (cid:0) D ( D ti ∪ D tj ) (cid:1) (cid:54) diam (cid:0) D ti (cid:1) + diam (cid:0) D tj (cid:1) + O (1) , (16)where the implicit constant depends on κ and ˆ D . This proves the second observation,and completes the proof of the lemma. (cid:3) The algorithm also admits the following extremal result, which says that thenumber of initial α -clusters in an α -covered droplet must be at least linear in thediameter of the droplet. It is precisely because of the existence of this result thatwe use the α -covering algorithm in the balanced setting, rather than the spanningalgorithm (Definition 6.13), for which there is no correspondingly strong extremallemma. Lemma 6.9 (Extremal lemma for α -covered droplets) . Let D be an α -covered drop-let. Then D ∩ A contains Ω (cid:0) diam( D ) (cid:1) disjoint α -clusters.Proof. The algorithm begins with k disjoint α -clusters, and ends with D = { D } .At each step of the algorithm the number of droplets is reduced by 1, and the sumof the diameters of the droplets increases by at most a constant, by (16). Hencediam( D ) (cid:54) k diam( ˆ D ) + O ( k ) , and so k = Ω (cid:0) diam( D ) (cid:1) , as required. (cid:3) It remains to show that if [ A ] = Z n , then there exists a large α -covered droplet.The main step is Lemma 6.11, below, which shows that an α -cover D of a set K is a reasonable approximation of the closure [ K ]. The basic idea is simple: sinceall α -clusters are contained in some droplet of D , the remaining ‘dust’ of infectedsites – that is, the set K \ ( D ∪ · · · ∪ D k ) – should contribute only locally to theset of eventually infected sites. We remark that a simplified version of the covering NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 35 algorithm was used in [13], not requiring Lemma 6.11, and in most cases resultingin non-optimal bounds.Before stating Lemma 6.11, let us note that we can replace half-planes by S B -droplets in the definition (13) of ρ . Lemma 6.10.
Let U be balanced, let D be an S B -droplet, and let Y ⊂ Z n have sizeat most α − . Then (cid:107) x − Y (cid:107) (cid:54) ρ for all x ∈ [ D ∪ Y ] \ D .Proof. Let { a u ∈ Z : u ∈ S B } be a collection of vectors such that D = (cid:92) u ∈S B H u ( a u ) , and let x ∈ [ D ∪ Y ] \ D . Since x / ∈ D , there exists u ∈ S B such that x / ∈ H u ( a u ).But (cid:12)(cid:12) Y \ H u ( a u ) (cid:12)(cid:12) (cid:54) α −
1, and x ∈ (cid:2) H u ( a u ) ∪ Y (cid:3) \ H u ( a u ) , so (cid:107) x − Y (cid:107) (cid:54) ρ by the definition of ρ . (cid:3) We are now ready to prove the key property of α -covers. Lemma 6.11.
Let U be a balanced update family, let K ⊂ Z be a finite set, let D = { D , . . . , D k } be an α -cover of K , and set Y := K \ (cid:0) D ∪ · · · ∪ D k (cid:1) . Then (cid:107) x − Y (cid:107) (cid:54) ρ for every x ∈ [ K ] \ (cid:0) D ∪ · · · ∪ D k (cid:1) .Proof. A first observation is that we may assume that α (cid:62)
2, since otherwise Y and[ K ] \ ( D ∪ · · · ∪ D k ) are empty (the latter because the covering algorithm stopped).We prove a slightly stronger statement: setting X = (cid:91) D ∈D D and Z = (cid:2) X ∪ Y (cid:3) , we shall show that the same conclusion holds with [ K ] replaced by Z .To begin, we partition Y into a collection Y , . . . , Y s of maximal strongly connectedcomponents, so in particular if y ∈ Y i and z ∈ Y j for some i (cid:54) = j , then (cid:107) y − z (cid:107) > κ = 2 ρ + ν. (17)(Note that the sets Y i are uniquely defined.) By the definition of an α -cover, wemust have | Y i | (cid:54) α −
1, and hence diam( Y i ) (cid:54) ( α − κ , for every i ∈ [ s ].For the clarity of what follows, we shall forget the labelling of the elements of D given in the statement of the lemma, so that we may reuse the notation D i . Sinceˆ D contains a ball of radius 2 ακ , we may assume that (cid:107) D − D (cid:48) (cid:107) > ακ (18)for every distinct pair D, D (cid:48) ∈ D . Thus, for each i ∈ [ s ] there is at most one droplet D i ∈ D such that Y i and D i are strongly connected, since κ + diam( Y i ) + κ (cid:54) ακ . (Of course we may have D i = D j for distinct i and j .) If there is no such D i thenset D i = ∅ . In particular, if D ∈ D and D (cid:54) = D i , then (cid:107) Y i − D (cid:107) > κ. (19)Now set Y (cid:48) i := [ D i ∪ Y i ] \ D i for each i ∈ [ s ], so that (cid:107) x − Y i (cid:107) (cid:54) ρ (20)for all x ∈ Y (cid:48) i and all i ∈ [ s ], by Lemma 6.10.We claim that X ∪ Y (cid:48) ∪ · · · ∪ Y (cid:48) s = Z ; (21)that is, that the set on the left-hand side is closed. Since the left-hand side of (21)may be re-written as X ∪ [ D ∪ Y ] ∪ · · · ∪ [ D s ∪ Y s ] , and each of these sets is closed individually (in the case of X this follows from (18)and the fact that κ > ν ), it is enough to show that if x ∈ Y (cid:48) i , and either y ∈ D forsome D ∈ D with D (cid:54) = D i , or y ∈ Y (cid:48) j with i (cid:54) = j , then x and y are not close enoughto interact; that is, (cid:107) x − y (cid:107) > ν . Indeed, if x ∈ Y (cid:48) i and y ∈ D for some D ∈ D with D (cid:54) = D i , then (cid:107) x − y (cid:107) (cid:62) (cid:107) Y i − D (cid:107) − (cid:107) x − Y i (cid:107) > κ − ρ = 2 ν by (19) and (20). On the other hand, if x ∈ Y (cid:48) i and y ∈ Y (cid:48) j , with i (cid:54) = j , then (cid:107) x − y (cid:107) (cid:62) (cid:107) Y i − Y j (cid:107) − (cid:107) x − Y i (cid:107) − (cid:107) y − Y i (cid:107) > κ − ρ = ν, by (17) and (20). Thus, (21) holds.We are now done, since we have shown that if x ∈ [ K ] \ X , then x ∈ Y (cid:48) i for some i ∈ [ s ], and we know that any such x satisfies (cid:107) x − Y i (cid:107) (cid:54) ρ . (cid:3) In order to prove the lower bound in Theorem 1.4 for balanced update families,we will in fact only need the following straightforward consequence of Lemma 6.11.
Lemma 6.12.
Let U be a balanced update family, and let A ⊂ Z n . If [ A ] = Z n ,then there exists an α -covered droplet D with log n (cid:54) diam( D ) (cid:54) n. (22) Proof.
Run the α -covering algorithm on Z n , with initial set A . If at some pointwe obtain an α -covered droplet D with diam( D ) (cid:62) log n , then choose the first suchdroplet, and observe that it satisfies (22), by the proof of Lemma 6.8. (Alternatively,choose any such droplet, and apply Lemma 6.8 to it.)So suppose that the α -covering algorithm stops without creating any droplets ofdiameter larger than log n , and let D = { D , . . . , D k } be the output of the algorithm.Setting Y := A \ (cid:0) D ∪ · · · ∪ D k (cid:1) , and applying Lemma 6.11, it follows that (cid:107) x − Y (cid:107) (cid:54) ρ for every x ∈ [ A ] \ (cid:0) D ∪ · · · ∪ D k (cid:1) . Since each strongly connected component of Y has size at most α −
1, and κ = 2 ρ + ν , it follows that different strongly connected NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 37 components of Y do not interact with one another. Recalling that (cid:107) D i − D j (cid:107) (cid:62) ακ for each i (cid:54) = j , it follows that [ A ] (cid:54) = Z n , which contradicts our assumption. (cid:3) The spanning algorithm: unbalanced families.
Next we describe our sec-ond analogue of the rectangles process, which will be a key tool in our analysis ofunbalanced models. Throughout this subsection we assume that U is unbalanced and that droplets are taken with respect to S U (so, in particular, D ( K ) now denotesthe smallest S U -droplet containing K ). We remind the reader that we define thealgorithm in Z to avoid some (unimportant) technical details relating to stronglyconnected sets of diameter Θ( n ).Recall from Section 2 that an S U -droplet D is said to be internally spanned by A if there exists a strongly connected set L ⊂ [ D ∩ A ] such that D ( L ) = D . Givena finite set K of infected sites, the output of the spanning algorithm is a minimalcollection D of internally spanned S U -droplets whose union contains K . At eachstep of the algorithm we maintain a partition K t = { K t , . . . , K tk } of K such thateach set [ K tj ] is strongly connected. Definition 6.13 ( The spanning algorithm ) . Let K = { x , . . . , x k } ⊂ Z be a setof infected sites. Set K := { K , . . . , K k } , where K j := { x j } for each 1 (cid:54) j (cid:54) k .Set t := 0, and repeat the following steps until STOP:1. If there are two sets K ti , K tj ∈ K t such that the set (cid:2) K ti (cid:3) ∪ (cid:2) K tj (cid:3) (23)is strongly connected, then set K t +1 := (cid:0) K t \ { K ti , K tj } (cid:1) ∪ (cid:8) K ti ∪ K tj (cid:9) , and set t := t + 1.2. Otherwise set T := t and STOP.The output of the algorithm is the span of K , (cid:104) K (cid:105) := (cid:8) D (cid:0) [ K T ] (cid:1) , . . . , D (cid:0) [ K Tk ] (cid:1)(cid:9) , where k = k − T .The following lemma provides an alternative description of the span of a set K . Lemma 6.14.
For every finite set K , we have (cid:104) K (cid:105) = (cid:8) D ( L ) , . . . , D ( L k ) (cid:9) , (24) where L , . . . , L k are the strongly connected components of [ K ] . This is not strictly speaking necessary: unlike in the previous subsection, the results here holdfor T -droplets for any T ⊂ S such that T -droplets are finite. Nevertheless, the only applicationsof the results in this subsection will be to unbalanced families, and it is useful to fix the set T . Proof.
We shall show that the sets [ K Ti ] are precisely the strongly connected com-ponents of [ K ]. Indeed, it follows from (23) (and a simple induction on t ) that [ K ti ]is strongly connected for every t ∈ [ T ] and 1 (cid:54) i (cid:54) k − t , and no two sets [ K Ti ]and [ K Tj ] are strongly connected, since the algorithm stopped at step T . Moreover,[ K ] = (cid:83) ki =1 [ K Ti ], since κ (from the definition of strongly connected) is greater than ν (the diameter of U ), and so no site can be infected by two or more of these sets. (cid:3) It is now easy to deduce that we can use the spanning algorithm to determinewhether or not D is internally spanned. Lemma 6.15. An S U -droplet D is internally spanned if and only if D ∈ (cid:104) D ∩ A (cid:105) .Proof. Applying Lemma 6.14 to K = D ∩ A , we see that D ∈ (cid:104) D ∩ A (cid:105) if and only if D ( L ) = D for some strongly connected component L of [ D ∩ A ]. But [ D ∩ A ] ⊂ D ,since S U ⊂ S , and so this is equivalent to the event that D is internally spanned. (cid:3) We can now prove the ‘Aizenman–Lebowitz lemma for internally spanned drop-lets’, which is the spanning analogue of Lemma 6.8 for α -covered droplets. For theapplications we shall need a slightly more general statement than before. Recallthat π ( D, u ) denotes the size of the projection of D in the direction u . We shallonce again use λ to denote a sufficiently large constant. Lemma 6.16.
Let D be an internally spanned S U -droplet, and let u ∈ S . Then forevery λ (cid:54) k (cid:54) π ( D, u ) , there exists an internally spanned S U -droplet D (cid:48) ⊂ D with k (cid:54) π ( D (cid:48) , u ) (cid:54) k. Proof.
Apply the spanning algorithm to K = D ∩ A and observe that, for every t (cid:54) T and every 1 (cid:54) i (cid:54) k − t , the droplet D ([ K ti ]) is internally spanned, since K ti ⊂ D ([ K ti ]) ∩ A and [ K ti ] is strongly connected.We claim that max (cid:8) π (cid:0) D ([ K ti ]) , u (cid:1) : K ti ∈ K t (cid:9) at most triples in size at each step, provided that this maximum is at least anabsolute constant. To see this, simply note that π (cid:0) D ( D ∪ D ) , u (cid:1) (cid:54) π ( D , u ) + π ( D , u ) + O (1) , (25)for any pair of droplets D and D that are within distance O (1) of one another, andthat D ([ Y ]) = D ( Y ) for any set Y , since S U ⊂ S . The lemma now follows easily, asin the proof of Lemma 6.8. (cid:3) We can now deduce an extremal lemma which, while much weaker than the cor-responding lemma for α -covered droplets (Lemma 6.9), is in fact tight up to theimplicit constant. This fact underlines how much we are ‘giving away’ in assumingonly that our droplets are spanned (rather than filled). Nevertheless, this lemmawill be sufficient to prove the base case (Lemma 6.18 below) of the main inductionargument (Lemma 8.3) for unbalanced models in Section 8. NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 39
Lemma 6.17. (Extremal lemma for internally spanned droplets.)
Let D be aninternally spanned S U -droplet. Then | D ∩ A | = Ω (cid:0) diam( D ) (cid:1) .Proof. As in the proof of the previous lemma, we apply the spanning algorithm with K = D ∩ A . The algorithm starts with k sets containing the individual elementsof D ∩ A , and it finishes with a collection (cid:104) D ∩ A (cid:105) = (cid:8) D (cid:0) [ K T ] (cid:1) , . . . , D (cid:0) [ K Tk ] (cid:1)(cid:9) such that D ∈ (cid:104) D ∩ A (cid:105) . At each step of the algorithm the number of sets in thecollection decreases by 1, and the sum of the diameters of the minimal dropletscontaining those sets increases by at most a constant, by (25). Hence,diam( D ) (cid:54) k (cid:88) i =1 diam (cid:0) D ([ K Ti ]) (cid:1) (cid:54) k diam (cid:0) D ([ K ]) (cid:1) + O ( k ) = O ( k ) , which implies that k = Ω (cid:0) diam( D ) (cid:1) , as required. (cid:3) Using Lemma 6.17, we can deduce a non-trivial bound on the probability that avery small droplet is internally spanned. As noted before, this will form the basecase of our induction argument in Lemma 8.3.Recall that I × ( D ) denotes the event that the S U -droplet D is internally spanned,and that w ( D ) and h ( D ) denote its width and height respectively, as defined inSection 2.5. Lemma 6.18.
For every η > , there exists δ > such that the following holds. Let D be an S U -droplet such that min (cid:8) w ( D ) , h ( D ) (cid:9) (cid:54) p − η . Then P p (cid:0) I × ( D ) (cid:1) (cid:54) p δ max { w ( D ) , h ( D ) } . Proof.
Let us write m ( D ) := min (cid:8) w ( D ) , h ( D ) (cid:9) and M ( D ) := max (cid:8) w ( D ) , h ( D ) (cid:9) .Suppose the S U -droplet D is internally spanned. Then by Lemma 6.17, D ∩ A mustcontain at least Ω (cid:0) M ( D ) (cid:1) sites. The probability that this occurs is at most (cid:18) O (cid:0) w ( D ) · h ( D ) (cid:1) δ (cid:48) · M ( D ) (cid:19) p δ (cid:48) M ( D ) (cid:54) (cid:0) O (1) · m ( D ) · p (cid:1) δ (cid:48) M ( D ) (cid:54) p δM ( D ) , for some δ, δ (cid:48) >
0, as required. (cid:3)
The final lemma of this subsection will be used in Section 8.1 as part of aninduction argument to prove the existence of ‘good and satisfied hierarchies’ forinternally spanned droplets (see the definitions and Lemma 8.7 contained withinthat section for details).
Lemma 6.19.
Let K ⊂ Z , with (cid:54) | K | < ∞ , be such that [ K ] is stronglyconnected. Then there exists a partition K = K ∪ K into non-empty (disjoint)sets such that [ K ] , [ K ] and [ K ] ∪ [ K ] are all strongly connected. Proof.
Run the spanning algorithm on K and consider the penultimate step. Since[ K ] is strongly connected, and therefore (cid:104) K (cid:105) = (cid:8) D (cid:0) [ K ] (cid:1)(cid:9) by Lemma 6.14, we have K T − = (cid:8) K , K (cid:9) for some K (cid:40) K and K (cid:40) K such that K = K ∪ K . By their construction inthe spanning algorithm, both [ K ] and [ K ] are strongly connected, and since K and K combine at the final step, so too is [ K ] ∪ [ K ]. (cid:3) The iceberg algorithm: unbalanced families with drift.
Our third al-gorithm will play a crucial role in the proof for update families that exhibit drift.Assume that U is unbalanced and let { u ∗ , − u ∗ } ⊂ S U be the pair of stable directionsgiven by Lemma 6.2 (and originally by Lemma 2.9), so in particularmin (cid:8) α ( u ∗ ) , α ( − u ∗ ) (cid:9) (cid:62) α + 1 . Recall from Section 2.3 that U exhibits drift if either of ¯ α ( u ∗ ) or ¯ α ( − u ∗ ) is infinite.We shall sometimes refer to u ∈ { u ∗ , − u ∗ } as a drift direction if ¯ α ( u ) = ∞ . Let usassume that α − ( u ∗ ) = ∞ , and observe that 1 (cid:54) α + ( u ∗ ) < ∞ .When our droplet is growing in direction u ∗ in a model with drift, it will tend toform a triangle, as in Section 5. In order to control the growth in this direction,we therefore need to ‘give away’ this triangle (in fact, a slightly larger one), andbound the growth outside it. The point of the algorithm defined in this subsectionis exactly to control this outwards growth using ‘icebergs’, defined as follows.Since α − ( u ∗ ) = ∞ , there exists a non-trivial interval [ u ∗ , u ] such that α − ( u ) = ∞ for every u ∈ [ u ∗ , u ], by Lemma 2.7. Fix such a u sufficiently close to u ∗ , in thefollowing sense: we choose u to be closer to u ∗ than any v ∈ S \ { u ∗ } perpendicularto x − y , where x, y ∈ (cid:83) X ∈U X ∪ { } and x (cid:54) = y . That we can choose such a u follows easily from the fact that U is a finite collection of finite sets. Finally, choose u ∈ ( u ∗ , u ) arbitrarily. Definition 6.20.
Let u ∈ ( u ∗ , u ]. A u -iceberg is any non-empty set J of the form J = (cid:0) H u ( a ) ∩ H u ∗ ( b ) (cid:1) \ H u , where a, b ∈ R . If X is a finite set of sites such that X (cid:54)⊂ H u , then denote by J u ( X )the smallest u -iceberg such that X ⊂ H u ∪ J u ( X ).Thus a u -iceberg is a discrete triangle whose sides are perpendicular to − u , u ∗ and u ; see Figure 5. The role of u is to ensure that that the angle between the( − u )-side and the u -side of a u -iceberg is uniformly bounded away from zero, whichwill be important in Lemma 6.25. We make a simple but key observation, whichfollows easily from the definition of u (cf. the proofs of Lemmas 2.7 and 3.5). If U does not exhibit drift then we shall not need the results proved in this section. NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 41 O (cid:0) γ/σ ( u ) (cid:1) O ( γ ) J uu ∗ u σ ( u ) Figure 5. A u -iceberg J , together with bounds on its width andheight given by Lemma 6.25. Lemma 6.21. If J is a u -iceberg, then H u ∪ J is closed.Proof. Suppose there exists z (cid:54)∈ H u ∪ J and a rule X ∈ U such that z + X ⊂ H u ∪ J .Since { u, u ∗ , u } ⊂ S , we cannot have X ⊂ H u or X ⊂ H u ( a ) ∩ H u ∗ ( b ). Hencethere exist x, y ∈ z + X with x (cid:54)∈ H u and y (cid:54)∈ H u ( a ) ∩ H u ∗ ( b ). But now x − y isperpendicular to a vector in the interval ( u ∗ , u ), contradicting our choice of u . (cid:3) We are now ready to introduce the iceberg algorithm, which is a modified versionof the covering algorithm allowing sites to be infected with the help of H u . At eachstep of the algorithm we have a collection W t of S U -droplets and u -icebergs; weeither take a droplet near H u and replace it by the smallest u -iceberg containingit, or we take two nearby sets in our collection, and replace them by either thesmallest u -iceberg containing their union (if they are sufficiently close to H u ), or bythe smallest droplet containing their union (otherwise). Definition 6.22 ( The u -iceberg algorithm ) . Let U be an unbalanced update familythat exhibits drift, and let u ∗ , u and u be as defined above, and let u ∈ ( u ∗ , u ].Suppose we are given: • K = (cid:8) x , . . . , x k (cid:9) ⊂ Z \ H u , a finite set of infected sites; • W = { W , . . . , W k } , a collection of copies of a fixed, sufficiently large S U -droplet ˆ D U , such that x j ∈ W j for each j = 1 , . . . , k .Set t := 0 and repeat the following steps until STOP:1. If there is a droplet W ti ∈ W t and an x ∈ Z such that the set W ti ∪ ( x + ˆ D U ) ∪ H u is strongly connected, then set W t +1 := (cid:0) W t \ { W ti } (cid:1) ∪ (cid:8) J u ( W ti ) (cid:9) , and set t := t + 1.
2. If not, but there are two sets W ti , W tj ∈ W t and an x ∈ Z such that the sets W ti ∪ W tj ∪ ( x + ˆ D U ) and W ti ∪ W tj ∪ ( x + ˆ D U ) ∪ H u are strongly connected, then set W t +1 := (cid:0) W t \ { W ti , W tj } (cid:1) ∪ (cid:8) J u ( W ti ∪ W tj ) (cid:9) , and set t := t + 1.3. If not, but there are two droplets W ti , W tj ∈ W t and an x ∈ Z such that theset W ti ∪ W tj ∪ ( x + ˆ D U )is strongly connected, then set W t +1 := (cid:0) W t \ { W ti , W tj } (cid:1) ∪ (cid:8) D ( W ti ∪ W tj ) (cid:9) , and set t := t + 1.4. Otherwise set T := t and STOP.The output of the algorithm is the family W := { W T , . . . , W Tk } . Definition 6.23.
Let u ∈ ( u ∗ , u ]. We say that W = { W , . . . , W k } is a u -icebergcover of a finite set K if W is a possible output of the u -iceberg algorithm withinput K . We say that an iceberg J is u -iceberg covered if W = { J } is a u -icebergcover of J ∩ A .Before continuing, let us note that u -iceberg covers are closed. Lemma 6.24.
Let u ∈ ( u ∗ , u ] , let K ⊂ Z be a finite set, and let W be an u -icebergcover of K . Then the set H u ∪ (cid:91) W ∈W W is closed under the U -bootstrap process.Proof. Since the algorithm has terminated, no two elements of W are strongly con-nected, and any element of W strongly connected to H u must be a u -iceberg. Thelemma now follows by Lemma 6.21. (cid:3) We can now prove our extremal result for icebergs; the lemma is illustrated inFigure 5. Let σ ( u ) denote the angle (in radians) between u and u ∗ . Lemma 6.25 (Extremal lemma for u -iceberg covers) . Let u ∈ ( u ∗ , u ] , let J be a u -iceberg covered u -iceberg, and let γ = | J ∩ A | . Then w ( J ) (cid:54) O (cid:0) γ/σ ( u ) (cid:1) and h ( J ) (cid:54) O ( γ ) , where the implicit constants may depend on U (and the fixed directions u ∗ , u and u ), but not on J , γ or u . NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 43
Proof.
Note first that T = O ( γ ) , (26)since at all but at most γ steps of the algorithm, |W t | is reduced by 1.Let D t and J t denote, respectively, the collections of droplets and u -icebergs in W t , so W t = D t ∪ J t . In order to prove the bound on the width of J in the lemma,we claim that, for each t (cid:54) T , (cid:88) D t ∈D t h ( D t ) + σ ( u ) (cid:88) J t ∈J t w ( J t ) = O ( t + γ ) = O ( γ ) . (27)The second equality is just (26). To see the first, note that the claim is clearly truewhen t = 0, and that at each step the left-hand side of (27) increases by at most O (1). Indeed, when two u -icebergs are replaced by another u -iceberg (as in step 2 ofthe u -iceberg algorithm), or when two droplets are replaced by another droplet (asin step 3 of the algorithm), this is clear, because the sums individually increase by atmost O (1) (as in (16)). When a droplet D t is replaced by a u -iceberg (as in step 1 ofthe algorithm), or a droplet D t and a u -iceberg are replace by a u -iceberg (again asin step 2 of the algorithm), the first sum in (27) decreases by h ( D t ), and the second(not including the factor of σ ( u )) increases by at most (cid:0) h ( D t ) + O (1) (cid:1) /σ ( u ). Thisproves (27), and hence, since the first sum is non-negative and the output of the u -iceberg algorithm is the single iceberg J , that σ ( u ) · w ( J ) = O ( γ ) , which is the first part of the lemma.For the second part, we claim that, for each t (cid:54) T , (cid:88) D t ∈D t (cid:0) σ ( u ) · w ( D t ) + h ( D t ) (cid:1) + (cid:88) J t ∈J t h ( J t ) = O ( t + γ ) = O ( γ ) . (28)As in the previous case, the second equality is just (26), and the claim is trivial when t = 0. Therefore it is enough to prove that the left-hand side of (28) increases by atmost O (1) at each step. When two droplets are replaced by another droplet or two u -icebergs are replaced by another u -iceberg, this is clear as before. On the otherhand, when a droplet D t is replaced by a u -iceberg, or a droplet D t and a u -icebergare replaced by a u -iceberg, the first sum decreases by σ ( u ) · w ( D t ) + h ( D t ) and thesecond sum increases by σ ( u ) · w ( D t ) + h ( D t ) + O (1). Thus we have h ( J ) = O ( γ ) , and this completes the proof of the lemma. (cid:3) The implicit constants here and subsequently may increase as the angle between u and u decreases, but this angle is bounded below by a function of u and u only, and the statement ofthe lemma permits such a dependence. The lower bound for balanced families
In this section we complete the proof of the lower bound in Theorem 1.4 forbalanced update families. The proof is a straightforward consequence of the α -covering algorithm, and the lemmas proved in Section 6.1. Theorem 7.1.
Let U be a balanced critical update family. Then p c ( Z n , U ) = Ω (cid:18) n (cid:19) /α Proof.
Let A be a p -random subset of Z n , where p = (cid:18) ε log n (cid:19) /α , for some sufficiently small constant ε = ε ( U ) >
0. We shall show that, with highprobability as n → ∞ , the U -bootstrap closure of A is not equal to Z n .Indeed, by Lemma 6.12, if [ A ] = Z n then there exists an α -covered droplet D withlog n (cid:54) diam( D ) (cid:54) n, and by Lemma 6.9 it follows that D ∩ A contains at least δ log n disjoint α -clusters,for some constant δ = δ ( U ) > D contains O (cid:0) diam( D ) (cid:1) = O (log n ) distinct α -clusters, it followsthat the probability D is α -covered is at most (cid:18) O (log n ) δ log n (cid:19) p αδ log n (cid:54) (cid:16) O (cid:0) p α log n (cid:1)(cid:17) δ log n (cid:54) n , since ε is sufficiently small. Finally, since there are at most n (log n ) O (1) choices ofthe droplet D having diameter at most 3 log n , it follows that P p (cid:0) [ A ] = Z n (cid:1) (cid:54) n · (log n ) O (1) · n = o (1) , as required. (cid:3) Note that we actually proved a stronger result than that stated in Theorem 7.1:it follows from the proof above that if ε = ε ( U ) > p = (cid:18) ε log n (cid:19) /α , and A is a p -random subset of Z n , then with high probability all strongly connectedcomponents of [ A ] on Z n have diameter O (log n ). NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 45 The lower bound for unbalanced families
In this section we shall prove the following theorem, and hence complete the proofof Theorem 1.4.
Theorem 8.1.
Let U be an unbalanced critical update family. Then p c ( Z n , U ) = Ω (cid:18) (log log n ) log n (cid:19) /α . Throughout the section we assume that U is unbalanced, and that droplets aretaken with respect to the set S U = { u ∗ , − u ∗ , u r , u l } as per Lemma 6.2, wheremin (cid:8) α ( u ∗ ) , α ( − u ∗ ) (cid:9) (cid:62) α + 1 and min (cid:8) ¯ α ( u l ) , ¯ α ( u r ) (cid:9) (cid:62) α, and u l and u r are contained in opposite semicircles separated by u ∗ and − u ∗ , with u l to the left and u r to the right of u ∗ . We also let ξ > δ (2 α + 1) defined below;see (31)), and we fix η := 110 α . (29)The main step in the proof of Theorem 8.1 is an upper bound on the probabilitythat a critical droplet is internally spanned. Recall from Definition 2.5 that in thissection a droplet D is said to be critical if its dimensions satisfy either( T ) w ( D ) (cid:54) p − α − / and ξp α log p (cid:54) h ( D ) (cid:54) ξp α log p , or( L ) p − α − / (cid:54) w ( D ) (cid:54) p − α − / and h ( D ) (cid:54) ξp α log p .The key bound we shall prove is that there exists δ > D is a criticaldroplet then P p (cid:0) I × ( D ) (cid:1) (cid:54) exp (cid:32) − δp α (cid:18) log 1 p (cid:19) (cid:33) . (30)(Recall again that I × ( D ) is the event that the S U -droplet D is internally spanned.)The proof of (30) will only be given towards the end of this section, in Lemma 8.33.We build up to that proof gradually via an induction argument, at each step ofwhich we bound the probability that droplets of certain (increasingly large) sizesare internally spanned.During the course of this section we shall use a large number of constants, withvarious dependencies. The main constants we shall use are δ ( β ), for 2 (cid:54) β (cid:54) α + 1,and the constants δ (which will appear in Lemma 8.33), ξ (from the definition of acritical droplet), and ε (which will be used in the proof of Theorem 8.1 in Section 8.6).These will be chosen so that1 (cid:29) δ (2) (cid:29) · · · (cid:29) δ (2 α + 1) (cid:29) ξ (cid:29) δ (cid:29) ε > , (31)by which we mean that the constants are chosen from left to right, so that each issufficiently small depending on all previous constants. Later in the section we shall introduce two further sequences of constants. The relationships between these newconstants and those in (31) will be set out explicitly in (38) and (39), below.Next we state the induction hypothesis. Definition 8.2.
For each β , β ∈ N with β + β (cid:54) α + 1, let IH( β , β ) be thefollowing statement: Let D be a droplet such that w ( D ) (cid:54) p − β (1 − η ) − η and h ( D ) (cid:54) p − β (1 − η ) − η . Then P p (cid:0) I × ( D ) (cid:1) (cid:54) p δ max { w ( D ) , h ( D ) } , (32)where δ = δ ( β + β ).We mention briefly that we would prefer the width and height conditions in Defini-tion 8.2 to be w ( D ) (cid:54) p − β (1 − η ) and h ( D ) (cid:54) p − β (1 − η ) respectively, but for technicalreasons we cannot quite square the bound on the width between β = 1 and β = 2;this is why the conditions take the slightly less elegant form above.The specific induction statements that we shall prove are:IH( β, β ) ⇒ IH( β + 1 , β ) for all 1 (cid:54) β (cid:54) α ;IH( β, β ) ⇒ IH( β, β + 1) for all 1 (cid:54) β (cid:54) α ; (cid:0) IH( β + 1 , β ) ∧ IH( β, β + 1) (cid:1) ⇒ IH( β + 1 , β + 1) for all 1 (cid:54) β (cid:54) α − . Note that IH(1 ,
1) is an immediate consequence of Lemma 6.18, and therefore to-gether these statements will be enough to prove the following lemma.
Lemma 8.3.
The assertions
IH( α + 1 , α ) and IH( α, α + 1) both hold.
Lemma 8.3 alone is not enough to give the bound (30) that we want on internallyspanned critical droplets. However, the techniques and lemmas that we use to proveLemma 8.3 will be the same as those that we use in the proof of Lemma 8.33 todeduce (30).The steps in the induction are of two types: horizontal steps of the formIH( β , β ) ⇒ IH( β + 1 , β ),and vertical steps of the formIH( β , β ) ⇒ IH( β , β + 1).Common to both is the key idea of crossings . Roughly speaking, these are eventsthat say that it is possible to ‘cross’ a parallelogram of sites from one side to the otherwith ‘help’ from one of the sides in the form of an infected half-plane. The eventsshould be thought of in the context of a growing droplet: a combination of crossingevents, one for each side of the droplet, enable an internally filled droplet to growinto a larger internally spanned droplet. We obtain bounds for the probabilitiesof crossings by showing that, to a certain level of precision, the most likely way NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 47 these events could occur is via the droplet (or half-plane) advancing row-by-row,rather than via the merging of many smaller droplets. One could think of this assaying that the growth mechanism we used to prove the upper bound for unbalancedfamilies in Theorem 5.1, which was indeed row-by-row, was essentially the ‘correct’mechanism. For vertical crossings in the case of models with drift, our proof willmake use of the results of Section 6.3 on the iceberg algorithm to bound the rangeof the U -bootstrap process in directions close to ± u ∗ . Full statements and proofs ofthe crossing lemmas, together with precise definitions, are given in Section 8.3.For the horizontal steps, in addition, we require the use of ‘hierarchies’ to boundthe extent of sideways growth at any given step. These are by now a standard toolin the bootstrap percolation literature, so we omit many of the details.There are six subsections in this section, which deal with the following aspectsof the proof: in the first we establish the hierarchies framework; in the second wederive a bound on the range of the U -bootstrap process in the geometry of the u -norm; in the third we prove the crossing lemmas; in the fourth we assemble thedifferent parts of the induction statement and prove Lemma 8.3; in the fifth wededuce Lemma 8.33, which is the bound for internally spanned critical droplets; andin the sixth and final subsection we complete the proof of Theorem 8.1.8.1. Hierarchies.
The use of hierarchies to control the formation of critical dropletswas introduced by Holroyd in [27] and has since developed into a standard techniquein the study of bootstrap percolation (see e.g. [3, 4, 18, 19, 25]). In this subsection werecall some of the standard definitions and lemmas, making only minor adaptationsalong the way to suit the general model. We are relatively brief with the details,referring the reader instead to [27], and the more recent refinements in [19, 25], fora more extensive introduction to the method.The key result of this subsection is Lemma 8.9, which gives an upper bound forthe probability that a droplet D is internally spanned in terms of the family ofhierarchies of D .Given a directed graph G and a vertex v ∈ V ( G ), we write N → G ( v ) for the set ofout-neighbours of v in G . Definition 8.4.
Let D be an S U -droplet. A hierarchy H for D is an ordered pair H = ( G H , D H ), where G H is a directed rooted tree such that all of its edges aredirected away from the root v root , and D H : V ( G H ) → Z is a function that assignsto each vertex of G H an S U -droplet, such that the following conditions are satisfied:(i) the root vertex corresponds to D , so D H ( v root ) = D ;(ii) each vertex has out-degree at most 2;(iii) if v ∈ N → G H ( u ) then D H ( v ) ⊂ D H ( u );(iv) if N → G H ( u ) = { v, w } then (cid:104) D H ( v ) ∪ D H ( w ) (cid:105) = { D H ( u ) } . Condition (iv) is equivalent to the statement that D H ( v ) ∪ D H ( w ) is stronglyconnected and that D H ( u ) is the smallest droplet containing their union. We shallusually abbreviate D H ( u ) to D u .The next definition controls the absolute and relative sizes of the droplets cor-responding to vertices of G H , which in turn allows us to control the number ofhierarchies. In order to limit the number of hierarchies as much as possible, wechoose the step size to be as large as possible, subject to the condition that we cancontrol the probability of each step. Definition 8.5.
Fix β ∈ N . A hierarchy H for an S U -droplet D is good if it satisfiesthe following conditions for each u ∈ V ( G H ):(v) u is a leaf if and only if w ( D u ) (cid:54) p − β (1 − η ) − η ;(vi) if N → G H ( u ) = { v } and | N → G H ( v ) | = 1 then p − β (1 − η ) − η / (cid:54) w ( D u ) − w ( D v ) (cid:54) p − β (1 − η ) − η ;(vii) if N → G H ( u ) = { v } and | N → G H ( v ) | (cid:54) = 1 then w ( D u ) − w ( D v ) (cid:54) p − β (1 − η ) − η ;(viii) if N → G H ( u ) = { v, w } then w ( D u ) − w ( D v ) (cid:62) p − β (1 − η ) − η / A of infected sites and to the U -bootstrap process. Given nested S U -droplets D ⊂ D (cid:48) , we write ∆( D, D (cid:48) ) for the event that D (cid:48) is internally spanned given that D is internally filled. That is,∆( D, D (cid:48) ) := (cid:8) D (cid:48) ∈ (cid:104) D ∪ ( D (cid:48) ∩ A ) (cid:105) (cid:9) . The final two conditions below ensure that a good hierarchy for an internally spanneddroplet D accurately represents the growth of the initial sites D ∩ A . Definition 8.6.
A hierarchy H for an S U -droplet D is satisfied by A if the followingevents all occur disjointly :(ix) if v is a leaf then D v is internally spanned by A ;(x) if N → G H ( u ) = { v } then ∆( D v , D u ) occurs.Having established all of the properties of hierarchies that we need, we now showthat there exists a good and satisfied hierarchy for every internally spanned droplet.The proof is almost identical to Propositions 31 and 33 of [27], which deal with the2-neighbour setting, except that here we use the spanning algorithm in place of therectangles process. We are therefore rather brief with the details. Lemma 8.7.
Let D be an S U -droplet internally spanned by A . Then there exists agood and satisfied hierarchy for D .Proof. In order to prove the lemma we consider a suitable ‘contraction’ of the treegiven by the spanning algorithm. To that end, let D = (cid:104) D ∩ A (cid:105) , and note that D ∈ D by Lemma 6.15, since D is internally spanned. The proof will be by induction on w ( D ), so note first that if w ( D ) (cid:54) p − β (1 − η ) − η then we may take V ( G H ) = { v root } . NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 49
For the induction step, first we claim that there exists a pair of sequences, D ∩ A ⊃ K ⊃ K ⊃ · · · ⊃ K m and D = D ⊃ D ⊃ · · · ⊃ D m , such that | K m | = 1 and such that for every 1 (cid:54) i (cid:54) m , D i = D ([ K i ]) and [ K i ] ∪ [ K i − \ K i ] is strongly connected . To construct these sequences, the idea is to run the spanning algorithm backwards,choosing at each step the larger of the two droplets. We make this idea precise usingLemma 6.19. Indeed, since D ∈ (cid:104) D ∩ A (cid:105) , there exists a set K ⊂ D ∩ A such that[ K ] is strongly connected and D = D ([ K ]). Now, given K i − such that [ K i − ] isstrongly connected, Lemma 6.19 gives a (non-trivial) partition K i ∪ K (cid:48) i of K i − suchthat [ K i ], [ K (cid:48) i ] and [ K i ] ∪ [ K (cid:48) i ] are all strongly connected. Set D i = D ([ K i ]) and D (cid:48) i = D ([ K (cid:48) i ]), where w ( D i ) (cid:62) w ( D (cid:48) i ).Now, let s (cid:62) w ( D s ) (cid:54) p − β (1 − η ) − η or w ( D ) − w ( D s ) (cid:62) p − β (1 − η ) − η , and attach a vertex u corresponding to D s to the root. If w ( D s ) (cid:54) p − β (1 − η ) − η and w ( D ) − w ( D s ) (cid:54) p − β (1 − η ) − η , then our construction of H is complete. If p − β (1 − η ) − η / (cid:54) w ( D ) − w ( D s ) (cid:54) p − β (1 − η ) − η , then we use the induction hypothesisto construct a good and satisfied (by K s ) hierarchy H (cid:48) for D s , and identify u withthe root of H (cid:48) . Finally, if w ( D ) − w ( D s ) (cid:62) p − β (1 − η ) − η then, by the minimality of s , we have w ( D s − ) − w ( D (cid:48) s ) (cid:62) w ( D s − ) − w ( D s ) (cid:62) p − β (1 − η ) − η . In this case we add a vertex v between u and the root, corresponding to D s − , andadd another vertex w attached to v , corresponding to D (cid:48) s . Now, using the inductionhypothesis, we construct good and satisfied (by K s and K s − \ K s respectively)hierarchies H (cid:48) and H (cid:48)(cid:48) for D s and D (cid:48) s , and identify u and w with the roots of H (cid:48) and H (cid:48)(cid:48) . It is straightforward to check that the hierarchies thus constructed satisfyconditions (i)–(x), as required. (cid:3) Remark 8.8.
We emphasize that the existence of a good and satisfied hierarchy for D does not imply that D is internally spanned, since the intersection of the events I × ( D v ) and ∆( D v , D u ) does not imply that D u is internally spanned, and since wedo not insist that [( D v ∪ D w ) ∩ A ] is strongly connected whenever N → G H ( u ) = { v, w } .The following fundamental bound on the probability that a droplet is inter-nally spanned (cf. [27, Section 10] or [4, Lemma 20]) will be used in the proofof Lemma 8.33 for type ( L ) critical droplets.Let us write H D for the set of all good hierarchies for D , and L ( H ) for the set ofleaves of G H . We will write (cid:80) u → v and (cid:81) u → v for the sum and product (respectively)over all pairs { u, v } ⊂ V ( G H ) such that N → G H ( u ) = { v } . Lemma 8.9.
Let D be an S U -droplet. Then P p (cid:0) I × ( D ) (cid:1) (cid:54) (cid:88) H∈H D (cid:18) (cid:89) u ∈ L ( H ) P p (cid:0) I × ( D u ) (cid:1)(cid:19)(cid:18) (cid:89) u → v P p (cid:0) ∆( D v , D u ) (cid:1)(cid:19) . (33) Proof. If D is internally spanned then by Lemma 8.7 there exists a good and satisfiedhierarchy for D . Taking the union bound over good hierarchies, and noting that fora fixed good hierarchy H the events I × ( D u ) (for u ∈ L ( H )) and ∆( D v , D u ) (for u → v ) are increasing and occur disjointly, the result follows from the van denBerg–Kesten inequality (Lemma 2.12). (cid:3) In order to use Lemma 8.9 we must bound the various probabilities that appear onthe right-hand side of (33), and the number of good hierarchies for D . A sufficientlystrong bound on P p (cid:0) I × ( D u ) (cid:1) (for each leaf u ∈ L ( H )) will follow immediately fromthe induction hypothesis; we will bound P p (cid:0) ∆( D v , D u ) (cid:1) in Section 8.3, again usingthe induction hypothesis, but this time the proof is considerably more difficult.Since our bounds will depend on w ( D u ) (for u ∈ L ( H )) and w ( D u ) − w ( D v ) (when N → G H ( u ) = { v } ), the following simple lemma will be useful. Lemma 8.10.
Let D be an S U -droplet, and H ∈ H D . Then (cid:88) u ∈ L ( H ) w ( D u ) + (cid:88) u → v (cid:16) w ( D u ) − w ( D v ) (cid:17) (cid:62) w ( D ) − O (cid:0) | V ( G H ) | (cid:1) , where the implicit constant depends only on U .Proof. This follows by combining Definition 8.4 with the geometric inequality w (cid:0) D ( D ∪ D ) (cid:1) (cid:54) w ( D ) + w ( D ) + O (1) , which holds for any pair of strongly connected droplets D and D (cf. (25)). (cid:3) Finally, to count the good hierarchies we partition the set H D according to thenumber of ‘big seeds’, as follows: Definition 8.11. If H is a hierarchy and v ∈ L ( H ), then we say that D v is a seed of H . If moreover w ( D v ) (cid:62) p − β (1 − η ) − η /
3, then we say that D v is a big seed of H . Remark 8.12.
The alert reader may have noticed that if H is good and has atleast two vertices, then all seeds of H are big. We will require only the followingslightly weaker fact: that every non-leaf of G H lies ‘above’ a big seed. This latterproperty holds for more general notions of a ‘good’ hierarchy (in particular, those inwhich the ‘step-size’ is much smaller than the maximum size of a seed), and playsan important role in some applications (see for example [25], where this methodwas first introduced). Since applying this more general method does not create anyadditional difficulties, we prefer to use this approach.Let us denote by b ( H ) the number of big seeds in a hierarchy H , by H bD the setof all good hierarchies for D that have exactly b big seeds, and by d ( H ) the depthof the tree G H , i.e., the maximum length of a path from the root to a leaf in G H . NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 51
Lemma 8.13. If D is an S U -droplet with h ( D ) = p − O (1) , then (cid:12)(cid:12) H bD (cid:12)(cid:12) (cid:54) exp (cid:20) O (cid:18) b · w ( D ) · p β (1 − η )+ η log 1 p (cid:19)(cid:21) Moreover, (cid:12)(cid:12) V ( G H ) (cid:12)(cid:12) = O (cid:16) b ( H ) · w ( D ) · p β (1 − η )+ η (cid:17) (34) for every H ∈ H D .Proof. By the definition of a good hierarchy, every vertex of G H that is not a leafmust lie above a big seed. This immediately implies that (cid:12)(cid:12) V ( G H ) (cid:12)(cid:12) (cid:54) · b ( H ) (cid:0) d ( H ) + 1 (cid:1) . We claim that either G H has only one vertex (in which case (cid:12)(cid:12) H bD (cid:12)(cid:12) = 1 and the lemmaholds trivially), or d ( H ) = O (cid:0) w ( D ) · p β (1 − η )+ η (cid:1) . Indeed, this follows from the fact that every two steps up G H , the width of thecorresponding droplet increases by Ω (cid:0) p − β (1 − η ) − η (cid:1) . We therefore have (34) for every H ∈ H D .Now, the number of choices for the tree G H is at most 2 O ( N ) , where N is ourbound on | V ( G H ) | . Moreover, for each u ∈ V ( G H ), there are at most p − O (1) possibledroplets D u . Hence (cid:12)(cid:12) H bD (cid:12)(cid:12) (cid:54) exp (cid:20) O (cid:18) b · w ( D ) · p β (1 − η )+ η log 1 p (cid:19)(cid:21) , as required. (cid:3) The range of unbalanced models with drift.
In this short section weassume that U is an unbalanced model with drift and we use the results about u -icebergs from Section 6.3 to prove a bound (see Lemma 8.15) on the range of the U -bootstrap process helped by a half-plane H u .Recall from Section 6.3 that if α − ( u ∗ ) = ∞ then we choose u ∈ S to the left ofand sufficiently close to u ∗ , so in particular α − ( u ) = ∞ for every u ∈ [ u ∗ , u ]. Wealso choose u ∈ ( u ∗ , u ). Similarly, if α + ( − u ∗ ) = ∞ then we choose a corresponding u (cid:48) ∈ S to the right of and sufficiently close to − u ∗ , and a corresponding u (cid:48) . Set S + U := (cid:40) [ u ∗ , u ] if α − ( u ∗ ) = ∞{ u ∗ } otherwise and S − U := (cid:40) [ u (cid:48) , − u ∗ ] if α + ( − u ∗ ) = ∞{− u ∗ } otherwise,and set S ± U := (cid:0) S + U ∪ S − U (cid:1) ∩ Q and S (cid:48) U := S ± U ∪ { u l , u r } . Recall also that for each u ∈ S + U , we defined σ ( u ) to be the angle between u and u ∗ ,and similarly for each u ∈ S − U . We define a norm (cid:107) · (cid:107) u on R as follows. Definition 8.14.
For each u ∈ S (cid:48) U , define (cid:107) x (cid:107) u := (cid:40) |(cid:104) x, u ∗ (cid:105)| + σ ( u ) |(cid:104) x, u ⊥ (cid:105)| if u ∈ S ± U \ { u ∗ , − u ∗ } , (cid:107) x (cid:107) if u ∈ { u ∗ , − u ∗ , u l , u r } , (35)where, as always, the unadorned norm (cid:107) · (cid:107) denotes the Euclidean norm on R .We record for later use the inequalities |(cid:104) x, u (cid:105)| (cid:54) (cid:107) x (cid:107) u (cid:54) · (cid:107) x (cid:107) , (36)which hold for every x ∈ R . Let ρ : S (cid:48) U × N → R be the function given by ρ ( u, γ ) := sup (cid:110)(cid:13)(cid:13) y − Y (cid:13)(cid:13) u : | Y | = γ − , y ∈ (cid:2) H u ∪ Y (cid:3) \ H u (cid:111) . (37)The key property that we need for the vertical crossings lemma, and the main resultof this section, is the following bound on ρ ( u, γ ), which is uniform in u . Lemma 8.15.
Let u ∈ S (cid:48) U and γ ∈ N , with γ (cid:54) ¯ α ( u ) . Then ρ ( u, γ ) = O ( γ ) , wherethe implicit constant depends only on U , and the fixed directions u ∗ , u and u .Proof. If u ∈ S U then ρ ( u, γ ) < ∞ follows from the proof of Lemma 6.5; indeed, theinduction step in that proof holds for all u ∈ Q and i < ¯ α ( u ). Since |S U | = 4 < ∞ ,we may therefore assume that u ∈ S ± U \ { u ∗ , − u ∗ } , and hence, by symmetry, that u ∈ S + U \ { u ∗ } . Note that this implies that α − ( u ∗ ) = ∞ .Let Y ⊂ Z be a set of size γ −
1, and let W be an u -iceberg cover of Y . The set H u ∪ (cid:91) W ∈W W contains Y and is closed, by Lemma 6.24. Hence, if y ∈ (cid:2) H u ∪ Y (cid:3) \ H u , then y ∈ W for some W ∈ W . But, by Lemma 6.25 and Definition 8.14, this implies that thereexists x ∈ W ∩ Y such that (cid:107) x − y (cid:107) u = O ( γ ), where the implicit constant dependsonly on U (and the fixed directions u ∗ , u and u ), as required. (cid:3) Crossing lemmas.
In this subsection we will bound the probabilities of certain‘crossing’ events, with a view to two specific applications. The horizontal crossingslemma (Lemma 8.18) will enable us to bound the probability of events of the form∆(
D, D (cid:48) ), which in turn allows us to bound the probability that ‘long’ droplets areinternally spanned using the hierarchies bound of Lemma 8.9. The vertical crossingslemma (Lemma 8.19) will enable us to bound (directly) the probability that ‘tall’droplets are internally spanned.Since there is significant overlap between the proofs for horizontal and verticalcrossings, it will be convenient to work in the following (slightly) more generalframework. If u ∈ { u ∗ , − u ∗ , u l , u r } then both inequalities are trivial. If u ∈ S ± U \ { u ∗ , − u ∗ } then note thatthe left-hand side is at most cos σ · |(cid:104) x, u ∗ (cid:105)| + sin σ · |(cid:104) x, u ⊥ (cid:105)| , which implies the first inequality, andthat σ ( u ) <
1, since u was chosen sufficiently close to u ∗ , which implies the second. NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 53
Definition 8.16.
Let u ∈ S (cid:48) U . A finite set is a u -strip if it is a T -droplet, where T = { u, − u, v, − v } and either • u ∈ { u l , u r } and v = u ∗ (a horizontal strip ), or • u ∈ S ± U = S (cid:48) U \ { u l , u r } and v = u ⊥ (a vertical strip ).Although it is convenient to define u -strips in terms of T -droplets, we stress againthat all sets described in this section as ‘droplets’ without reference to a set T areassumed to be S U -droplets.Recall from (14) that κ = 3 ν when U is unbalanced, that we denote by G κ thegraph with vertex set Z and edge set E = (cid:8) xy : (cid:107) x − y (cid:107) (cid:54) κ (cid:9) , and that a stronglyconnected component is defined to be a component in this graph. Recall also thatthe u -projection π ( K, u ) of a finite set K ⊂ Z was defined in (3) by π ( K, u ) = max (cid:8) (cid:104) x − y, u (cid:105) : x, y ∈ K (cid:9) , and that if D is a T -droplet and u ∈ T , then the u -side ∂ ( D, u ) of D was definedin (5) to be the set D ∩ (cid:96) u ( i ), where i is maximal so that this set is non-empty. Definition 8.17.
Let u ∈ S (cid:48) U , let S be a u -strip, and let x ∈ ∂ ( S, − u ). We saythat S is u -crossed if there exists a strongly connected set in [ H u ( x ) ∪ ( S ∩ A )] thatintersects both H u ( x ) and ∂ ( S, u ).Note that the half-plane H u ( x ) does not depend on the choice of x ∈ ∂ ( S, − u ).Unless the precise position of the u -strip is important, we will usually assume thatthe ( − u )-side of the u -strip is a subset of (cid:96) u .Before continuing with the results of this subsection, we give a more completeaccount of the relationships between the different constants of this section thanthat given in (31). We mentioned that, during the course of the inductive proof ofLemma 8.3, two sequences of constants would be defined, in addition to the constants δ ( β ) already introduced in Definition 8.2. These sequences are δ (cid:48) (2) , . . . , δ (cid:48) (2 α + 1),which appear in the statements of Lemmas 8.18 and 8.19, and κ (2) , . . . , κ (2 α + 1),which appear in Definition 8.20. These constants will be chosen to have the followingrelative sizes. First, for each 2 (cid:54) β (cid:54) α ,1 (cid:29) δ ( β ) (cid:29) κ ( β ) (cid:29) δ (cid:48) ( β ) (cid:29) δ ( β + 1) > , (38)and second, δ (2 α + 1) (cid:29) κ (2 α + 1) (cid:29) δ (cid:48) (2 α + 1) (cid:29) ξ (cid:29) δ (cid:29) ε > . (39)We emphasize again that these statements mean that the constants are chosen fromleft to right, and that each is chosen to be sufficiently small depending on all previ-ously chosen constants. Note that these two sets of relations subsume those in (31).The main results of this subsection are Lemmas 8.18 and 8.19, below. One maythink of the lemmas as exchanging bounds on the probability that a droplet isinternally spanned for bounds on the probability that similarly sized u -strips are u -crossed, for some u . (It may be helpful, therefore, to think of the δ (cid:48) ( β ) as beingto crossing u -strips as the δ ( β ) are to internally spanning S U -droplets.)The first of the two lemmas bounds the probability of horizontal crossings. Lemma 8.18.
Let S be a u -strip, where u ∈ { u l , u r } .(i) Let (cid:54) β (cid:54) α and (cid:54) β (cid:54) α , and suppose that IH( β , β ) holds. If ξ − (cid:54) π ( S, u ) (cid:54) p − β (1 − η ) − η and h ( S ) (cid:54) p − β (1 − η ) − η , then S is u -crossed with probability at most p δ (cid:48) π ( S,u ) , where δ (cid:48) = δ (cid:48) ( β + β ) .(ii) Suppose that IH( α, α + 1) holds. If (cid:54) π ( S, u ) (cid:54) p − α (1 − η ) − η and h ( S ) (cid:54) ξp α log 1 p , then S is u -crossed with probability at most π ( S, u ) · exp (cid:16) − p O ( ξ ) · π ( S, u ) (cid:17) , where the implicit constant depends on κ (2 α + 1) . The second of our two main crossing lemmas deals with vertical crossings. Recallthat u ∗ has difficulty at least α + 1, and therefore either ¯ α ( u ∗ ) (cid:62) α + 1 or α − ( u ∗ ) = ∞ . The behaviour of the U -bootstrap process differs markedly depending on whichof these two cases we are in. Note that, while the lemma is stated only for u ∈ S + U ,it is plain by symmetry that a similar statement holds for u ∈ S − U . Moreover, forsimplicity we assume that p is chosen so that if σ ( u ) = p − η , then u ∈ Q . Lemma 8.19.
Let u ∈ S + U ∩ Q be such that either u = u ∗ and ¯ α ( u ∗ ) (cid:62) α + 1 , or σ ( u ) = p − η and α − ( u ∗ ) = ∞ . Let (cid:54) β (cid:54) α + 1 and (cid:54) β (cid:54) α , and suppose that IH( β , β ) holds. If S is a u -strip with w ( S ) (cid:54) p − β (1 − η ) − η , h ( S ) (cid:54) p − β (1 − η ) − η , and π ( S, u ) (cid:62) ξ − , then S is u -crossed with probability at most p δ (cid:48) π ( S,u ) , where δ (cid:48) = δ (cid:48) ( β + β ) . Observe that if u ∗ is not a drift direction (that is, if α − ( u ∗ ) < ∞ ) then the lemmasays it is unlikely that a u ∗ -strip of an appropriate size is u ∗ -crossed – this is whatone would expect. If u ∗ is a drift direction, on the other hand, then instead thelemma is stated in terms of crossing u -strips, where σ ( u ) = p − η . Why might thisbe the natural direction in which to bound growth? Since u ∗ is a drift direction, α + ( u ∗ ) may be as small as 1, and therefore one would expect a triangle of sites ofslope p to form on the u ∗ -side of the droplet, similarly to the set T in Figure 4.By rotating u ∗ through an angle of p − η , we are ‘giving away’ more sites than onewould expect to become infected, but not so many more that it adversely affects thebound. We expand on these remarks before the proof of the lemma. NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 55
The first step towards proving Lemmas 8.18 and 8.19 is a deterministic descriptionof the structure of S ∩ A when S is u -crossed, which is given by Lemma 8.22. Wepartition the u -strip into consecutive u -strips S , . . . , S m of constant u -projection,and we consider how the infection could spread from the ( − u )-side of S (and theadjacent half-plane H u ) to the u side of S . One of the key concepts we use will bethat of being ‘ u -weakly connected’, defined as follows. Definition 8.20.
Fix β , β ∈ N with β + β (cid:54) α + 1, and let κ = κ ( β + β ).For each u ∈ S (cid:48) U and γ ∈ N , we say that:( a ) A set Z ⊂ Z is u -weakly connected if it is connected in the graph G u,κ withvertex set Z and edge set E ( G u,κ ) = (cid:8) xy : (cid:107) x − y (cid:107) u (cid:54) κ (cid:9) .( b ) A u -weak γ -cluster is a set of γ sites that is u -weakly connected.Note that we suppress the dependence on the pair ( β , β ) in the definition of a u -weak γ -cluster; we trust that this will not cause any confusion. We remark that inwhat follows we will always take γ (cid:54) ¯ α ( u ), so if Y is a u -weak ( γ − H u ∪ Y only causes ‘local’ new infections,measured in the u -norm.We can now define the deterministic structural property that we shall prove (inLemma 8.22, below) is implied by the event that S is u -crossed by A . The definitionis illustrated in Figure 6. Definition 8.21.
Fix β , β ∈ N with β + β (cid:54) α + 1, and let κ = κ ( β + β ).Let u ∈ S (cid:48) U and γ ∈ N , and suppose that S is a u -strip. Let S ∪ · · · ∪ S m +1 be apartition of S into u -strips, with S i adjacent to S i +1 for each i ∈ [ m ],3 κ γ (cid:54) π ( S i , u ) = π ( S j , u ) (cid:54) κ γ for each i, j ∈ [ m ], and π ( S m +1 , u ) < κ γ .A ( u, γ ) -partition for S ∩ A is a sequence ( a , . . . , a k ) of positive integers with a + · · · + a k = m , such that for each 1 (cid:54) j (cid:54) k , setting t j = a + · · · + a j , either • a j = 1 and S t j ∩ A contains a u -weak γ -cluster, or • there exists an S U -droplet D internally spanned by (cid:0) S t j − +1 ∪ · · · ∪ S t j (cid:1) ∩ A ,where max (cid:8) w ( D ) , h ( D ) (cid:9) (cid:62) a j κ . In the following lemma we need an upper bound on γ when ¯ α ( u ) = ∞ . For thispurpose, fix λ := 5 α/η = 50 α (40)throughout this section, and observe that κ (2) (cid:29) ρ ( u, γ ) (41)for every u ∈ S (cid:48) U and γ ∈ N such that γ (cid:54) min { ¯ α ( u ) , λ } . Indeed, ρ ( u, γ ) is boundedabove by a constant that depends only on U and α = α ( U ), by Lemma 8.15 and (29),and κ (2) was chosen in (38) to be sufficiently large (depending on U ). γ γ γ γD u r H u r Figure 6. A u r -crossed u r -strip S together with a possible ( u r , γ )-partition for S ∩ A in which a = a = a = a = 1 and a = 3.The following deterministic lemma, which says that every u -crossed strip has a( u, γ )-partition, is the key step in the proof of Lemmas 8.18 and 8.19. Lemma 8.22.
Let β , β ∈ N with β + β (cid:54) α + 1 , and set κ = κ ( β + β ) .Suppose that u ∈ S (cid:48) U and γ ∈ N satisfy γ (cid:54) min (cid:8) ¯ α ( u ) , λ (cid:9) , and let S be a u -strip.If S is u -crossed by A , then there exists a ( u, γ ) -partition for S ∩ A . Roughly speaking, the proof of the lemma is as follows. We shall show that if S ∩ A does not contain a u -weak γ -cluster, then S cannot itself be u -crossed. Since S is u -crossed, this will allow us to deduce that there exists a droplet D internallyspanned by S ∩ A such that D ∩ S (cid:54) = ∅ , and moreover such that D extends at leasthalfway across S . We call such a droplet D a saver . Letting a be maximal suchthat D ∩ S a (cid:54) = ∅ , the result follows by induction on m . Proof of Lemma 8.22.
As noted above, the proof is by induction on m . If m = 0there is nothing to prove, so let m (cid:62) m . If S ∩ A contains a u -weak γ -cluster then we aredone, since we may set a = 1 and observe that S \ S is u -crossed by A .So assume that S ∩ A does not contain a u -weak γ -cluster, let Y , . . . , Y s be thecollection of u -weakly connected components in S ∩ A that are each also u -weaklyconnected to H u , and set Y := Y ∪ · · · ∪ Y s and Z := [ H u ∪ Y ] \ H u . We claim that | Y i | (cid:54) γ − (cid:54) i (cid:54) s . Indeed, if | Y i | (cid:62) γ then there existsa u -weak γ -cluster Y (cid:48) ⊂ Y i such that (cid:107) y − H u (cid:107) u (cid:54) κ γ for every y ∈ Y (cid:48) . Recallingfrom (36) that (cid:104) x, u (cid:105) (cid:54) (cid:107) x (cid:107) u for every x ∈ Z , and that π ( S , u ) (cid:62) κ γ , it followsthat Y (cid:48) ⊂ S . This contradicts our assumption that S ∩ A does not contain a u -weak γ -cluster, and thus proves that | Y i | (cid:54) γ − (cid:54) i (cid:54) s , as claimed. NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 57
We next claim that H u ∪ Z = [ H u ∪ Y ] ∪ · · · ∪ [ H u ∪ Y s ] . (42)To prove this, let z i ∈ [ H u ∪ Y i ] \ H u and z j ∈ [ H u ∪ Y j ] \ H u , and note that (cid:107) z i − Y i (cid:107) u (cid:54) ρ ( u, γ ) and (cid:107) z j − Y j (cid:107) u (cid:54) ρ ( u, γ ), by the definition of ρ ( u, γ ). Hence (cid:107) z i − z j (cid:107) (cid:62) (cid:107) z i − z j (cid:107) u (cid:62) κ − ρ ( u, γ )2 > ν, where the first inequality follows from (36), the second by the triangle inequality,and the third from (41), since γ (cid:54) min { ¯ α ( u ) , λ } . Therefore, the set[ H u ∪ Y ] ∪ · · · ∪ [ H u ∪ Y s ]is closed (and contains Y ), which proves (42). Note that it follows from the aboveargument that moreover (cid:107) z − Y (cid:107) u (cid:54) ρ ( u, γ ) (43)for every z ∈ Z .We are now ready to prove our key claim, which says that, under our assumptionthat S does not contain a u -weak γ -cluster, there exists a droplet that is internallyspanned by S ∩ A , and has large intersection with S . Claim 8.23.
There exists a droplet D internally spanned by S ∩ A such that |(cid:104) D − H u , u (cid:105)| (cid:54) κ γ and max (cid:8) w ( D ) , h ( D ) (cid:9) (cid:62) κ , where (cid:104) D − H u , u (cid:105) = min (cid:8) (cid:104) x − y, u (cid:105) : x ∈ D, y ∈ H u (cid:9) .Proof of Claim 8.23. The first step is to show that there exist z ∈ H u ∪ Z and w ∈ [ S ∩ A \ Y ] with (cid:107) w − z (cid:107) u (cid:54) · (cid:107) w − z (cid:107) (cid:54) κ. (44)Note that the first inequality follows by (36), so we just need to prove the second.To do so, recall that S is u -crossed by A , which means that there exists a stronglyconnected component L ⊂ [ H u ∪ ( S ∩ A )] that intersects both H u and ∂ ( S, u ). Notethat H u ∪ Z does not intersect ∂ ( S, u ), by (43), and since m (cid:62) π ( S, u ) (cid:62) κ γ . Now, either [ H u ∪ ( S ∩ A )] = [ H u ∪ Y ] ∪ [ S ∩ A \ Y ], in which casethere must exist z ∈ [ H u ∪ Y ] = H u ∪ Z and w ∈ [ S ∩ A \ Y ] with (cid:107) w − z (cid:107) (cid:54) κ , or[ H u ∪ ( S ∩ A )] (cid:54) = [ H u ∪ Y ] ∪ [ S ∩ A \ Y ], in which case there must exist z ∈ H u ∪ Z and w ∈ [ S ∩ A \ Y ] with (cid:107) w − z (cid:107) (cid:54) ν . Since κ = 3 ν , in either case (44) holds.Now, let D be the output of the spanning algorithm with input S ∩ A \ Y , and let D ∈ D be the droplet spanned by the strongly connected component of [ S ∩ A \ Y ]containing w . If z ∈ H u , then it follows by (44) and the u -norm bound in (36) that (cid:12)(cid:12) (cid:104) D − H u , u (cid:105) (cid:12)(cid:12) (cid:54) |(cid:104) w − z, u (cid:105)| (cid:54) (cid:107) w − z (cid:107) u (cid:54) κ (cid:54) γκ . xwzy Z D S H u Figure 7.
The situation in the proof of Claim 8.23 is depicted as-suming z ∈ Z . The size of the projection (cid:12)(cid:12) (cid:104) D − H u , u (cid:105) (cid:12)(cid:12) is at most thetotal length of the dashed line in the u -norm.On the other hand, if z ∈ Z then (cid:107) z − Y (cid:107) u (cid:54) ρ ( u, γ ) (cid:28) κ , by (41) and (43).Therefore, recalling that every y ∈ Y is within distance at most γκ of H u in the u -norm, the triangle inequality and (36) gives |(cid:104) D − H u , u (cid:105)| (cid:54) (cid:107) w − z (cid:107) u + (cid:107) z − Y (cid:107) u + γκ (cid:54) γκ , as required.To bound the dimensions of D , let x ∈ D ∩ A \ Y , and observe that (cid:13)(cid:13) x − ( H u ∪ Y ) (cid:13)(cid:13) u > κ by the definition of Y . Using (43), it follows that (cid:13)(cid:13) x − ( H u ∪ Z ) (cid:13)(cid:13) u > κ − ρ ( u, γ ) , and hence, by (36), (cid:13)(cid:13) x − ( H u ∪ Z ) (cid:13)(cid:13) (cid:62) (cid:13)(cid:13) x − ( H u ∪ Z ) (cid:13)(cid:13) u (cid:62) κ − ρ ( u, γ )2 . However, by (44) and our choice of w , we also have (cid:13)(cid:13) w − ( H u ∪ Z ) (cid:13)(cid:13) (cid:54) (cid:107) w − z (cid:107) (cid:54) κ. NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 59
Since x, w ∈ D , it follows thatmax (cid:8) w ( D ) , h ( D ) (cid:9) (cid:62) (cid:107) w − x (cid:107) (cid:62) (cid:13)(cid:13) x − ( H u ∪ Z ) (cid:13)(cid:13) − (cid:13)(cid:13) w − ( H u ∪ Z ) (cid:13)(cid:13) (cid:62) κ − ρ ( u, γ ) − κ (cid:62) κ , by the triangle inequality and (41), as required. (cid:3) To complete the proof of the lemma, simply set a = max { i : D ∩ S i (cid:54) = ∅} ,and observe that S \ ( S ∪ · · · ∪ S a ) is u -crossed by A . It follows easily fromClaim 8.23, our choice of a , and the fact that π ( S i , u ) (cid:62) κ γ for every i ∈ [ m ],that max { w ( D ) , h ( D ) } (cid:62) a κ /
5, as required. (cid:3)
We next prove an upper bound, depending on u ∈ S (cid:48) U and on the size of S , onthe probability that a p -random subset of a u -strip S admits a ( u, γ )-partition. Inorder to simplify the statement, given u ∈ S (cid:48) U and a u -strip S , let g u ( S ) denote thenumber of u -weak γ -clusters in a sub-strip S (cid:48) ⊂ S of u -projection 4 κ γ . Lemma 8.24.
Let β , β ∈ N with β + β (cid:54) α + 1 , set κ = κ ( β + β ) , andassume that IH( β , β ) holds. Let u ∈ S (cid:48) U and γ ∈ N , with γ (cid:54) min (cid:8) ¯ α ( u ) , λ (cid:9) , andlet S be a u -strip with | S | (cid:54) p − α , and ξ − (cid:54) π ( S, u ) (cid:54) p − β (1 − η ) − η , (45) where β := min { β , β } . Then the probability that S ∩ A admits a ( u, γ ) -partition isat most π ( S, u ) · max (cid:54) j (cid:54) m (cid:16) − (cid:0) − p γ (cid:1) g u ( S ) (cid:17) m − j (cid:0) π ( S, u ) · p α (cid:1) j , (46) where m = (cid:98) π ( S, u ) / κ γ (cid:99) .Proof. We first deal with a technicality: the saver droplets need only be internallyspanned by the sites in S ∩ A ; they do not have to be contained in S , and thereforetheir dimensions may be too large to use IH( β , β ). Moreover, even if the savers are contained in S , they may still have dimensions too large to use IH( β , β ). However,neither of these is a problem, as we now show. Let D be any saver droplet (so D isinternally spanned by S ∩ A ) such that either w ( D ) (cid:62) p − β (1 − η ) − η or h ( D ) (cid:62) p − β (1 − η ) − η . (47)Then by Lemma 6.16, applied once with u = u ∗ and again if necessary with u = u ⊥ ,there exists a droplet D (cid:48) ⊂ D , also spanned by sites in S ∩ A , such that either w ( D (cid:48) ) (cid:54) p − β (1 − η ) − η and p − β (1 − η ) − η / (cid:54) h ( D (cid:48) ) (cid:54) p − β (1 − η ) − η , or p − β (1 − η ) − η / (cid:54) w ( D (cid:48) ) (cid:54) p − β (1 − η ) − η and h ( D (cid:48) ) (cid:54) p − β (1 − η ) − η . Therefore, by IH( β , β ), we have P p (cid:0) I × ( D (cid:48) ) (cid:1) (cid:54) p δk/ , where δ = δ ( β + β ) and k := min (cid:8) p − β (1 − η ) − η , p − β (1 − η ) − η (cid:9) . But k (cid:62) π ( S, u ), since S satisfies (45), and therefore P p (cid:0) I × ( D (cid:48) ) (cid:1) (cid:54) p δπ ( S,u ) / . Hence, since there are at most p − α distinct S U -droplets spanned by sites in S , itfollows that the probability S admits a ( u, γ )-partition containing a saver droplet D satisfying (47) is at most p − α · p δπ ( S,u ) / . Now, recalling from (38) and (39) that π ( S, u ) (cid:62) ξ − (cid:29) κ ( β + β ) (cid:29) δ ( β + β ) − , it follows that p δ ( β + β ) π ( S,u ) / − α (cid:54) p δ ( β + β ) π ( S,u ) / (cid:54) p αm , and this is at most (46) (with j = m ).Let us therefore assume from now on that if D is a saver droplet in a ( u, γ )-partition of S , then the dimensions of D satisfy w ( D ) (cid:54) p − β (1 − η ) − η and h ( D ) (cid:54) p − β (1 − η ) − η . (48)Let S ∪ · · · ∪ S m (cid:48) +1 be a partition of S into u -strips as in Definition 8.21, and notethat we have m (cid:48) (cid:62) m , since (cid:107) S i − S i +1 (cid:107) (cid:54) i ∈ [ m (cid:48) ]. Note also that m (cid:62) π ( S, u ) (cid:62) ξ − (cid:29) κ ( β + β ), and γ (cid:54) λ .Next, observe that for each 1 (cid:54) i (cid:54) m , the probability that S i ∩ A contains a u -weak γ -cluster is at most 1 − (cid:0) − p γ (cid:1) g u ( S ) , (49)by Harris’s inequality, since by definition there are at most g u ( S ) such sets in S i .Now, as noted above, there are at most p − α distinct S U -droplets that are inter-nally spanned by a subset of S , and by IH( β , β ) and (48), each such droplet D isinternally spanned with probability at most p δ ( β + β ) max { w ( D ) ,h ( D ) } . Thus, for each a ∈ [ m ] and 0 (cid:54) t (cid:54) m − a , the probability that there is a droplet D with max { w ( D ) , h ( D ) } (cid:62) aκ ( β + β ) / D is internally spanned by( S t +1 ∪ · · · ∪ S t + a ) ∩ A is at most p δ ( β + β ) κ ( β + β ) a/ − α (cid:54) p αa , (50)since κ ( β + β ) (cid:29) δ ( β + β ) − .Finally, note that there are at most π ( S, u ) j partitions of m (cid:48) containing at least m (cid:48) − j ones. By (49) and (50), and taking a union bound over j , it follows that S admits a ( u, γ )-partition with probability at most π ( S, u ) · max (cid:54) j (cid:54) m (cid:16) − (cid:0) − p γ (cid:1) g u ( S ) (cid:17) m − j (cid:0) π ( S, u ) · p α (cid:1) j , Recall that D might not be contained in S ; however, any S U -droplet spanned by sites in S is contained in D ( S ) (the smallest S U -droplet that contains S ), which has size O ( | S | ). Also, an S U -droplet is determined uniquely by two opposite corners, for each of which there are at most O ( | D ( S ) | ) choices. NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 61 as claimed. (cid:3)
We shall now apply Lemmas 8.22 and 8.24 three times: once to prove Lemma 8.18for horizontal crossings, and twice to prove Lemma 8.19 for vertical crossings, onceeach for drift and non-drift directions. We begin with horizontal crossings.
Proof of Lemma 8.18.
Suppose first that IH( β , β ) holds, where 1 (cid:54) β (cid:54) α and1 (cid:54) β (cid:54) α , and let S be a u -strip, where u ∈ { u l , u r } and ξ − (cid:54) π ( S, u ) (cid:54) p − β (1 − η ) − η and h ( S ) (cid:54) p − β (1 − η ) − η . If S is u -crossed by A , then, recalling that ¯ α ( u ) (cid:62) α , it follows by Lemma 8.22 thatthere exists a ( u, α )-partition for S ∩ A .Let S ∪ · · · ∪ S m (cid:48) +1 be a partition of S into u -strips as in Definition 8.21, and notethat there are at most O (cid:0) h ( S ) (cid:1) u -weak α -clusters in each sub-strip S i , where theimplicit constant depends on κ = κ ( β + β ). It therefore follows from Lemma 8.24that S ∩ A admits a ( u, α )-partition with probability at most π ( S, u ) · max (cid:54) j (cid:54) m (cid:16) − (cid:0) − p α (cid:1) O ( h ( S )) (cid:17) m − j (cid:0) π ( S, u ) · p α (cid:1) j , (51)where m = (cid:98) π ( S, u ) / κ α (cid:99) . Since h ( S ) (cid:54) p − β (1 − η ) − η and 1 (cid:54) β (cid:54) α , we have − (1 − p α ) O ( h ( S )) = O (cid:0) p α − β (1 − η ) − η (cid:1) (cid:54) p η . Also, since π ( S, u ) (cid:54) p − β (1 − η ) − η and 1 (cid:54) β (cid:54) α , we have π ( S, u ) · p α (cid:54) p η .Therefore, recalling that m (cid:29) π ( S, u ) (cid:62) ξ − (cid:29) κ ( β + β )), it followsthat (51) is at most π ( S, u ) · max (cid:54) j (cid:54) m p η ( m − j ) · p ηj = π ( S, u ) · p ηm (cid:54) p δ (cid:48) π ( S,u ) , as required, where δ (cid:48) = δ (cid:48) ( β + β ) (cid:54) η/ κ ( β + β ) α .Now suppose that IH( α, α + 1) holds, and let S be a u -strip, where u ∈ { u l , u r } and 1 (cid:54) π ( S, u ) (cid:54) p − α (1 − η ) − η and h ( S ) (cid:54) ξp α log 1 p . Then, exactly as above, it follows that (51) is an upper bound on the probabilitythat S is u -crossed by A . Since h ( S ) (cid:54) ξp α log p , we have1 − (1 − p α ) O ( h ( S )) (cid:54) − exp (cid:16) − O (cid:0) p α · h ( S ) (cid:1)(cid:17) (cid:54) − p O ( ξ ) (cid:54) e − p O ( ξ ) , where the implicit constant depends on κ (2 α + 1). Also, since π ( S, u ) (cid:54) p − α (1 − η ) − η ,we have π ( S, u ) · p α (cid:54) p η , as before. Since π ( S, u ) = Θ( m ), it follows that (51) isat most π ( S, u ) · max (cid:54) j (cid:54) m (cid:16) e − p O ( ξ ) (cid:17) m − j p ηj (cid:54) π ( S, u ) · exp (cid:16) − p O ( ξ ) · π ( S, u ) (cid:17) , where the implicit constants depend on κ (2 α + 1), as required. (cid:3) Here, and below, we use the inequality 1 − ax (cid:54) (1 − x ) a , which is valid if x (cid:54) a (cid:62) Before proving Lemma 8.19, which bounds the probability of a vertical crossing,let us first note the following bounds on the probability of the event ∆(
D, D (cid:48) ), whichhold when w ( D (cid:48) ) − w ( D ) is not too large, and follow easily from Lemma 8.18. Wewill use these bounds, together with Lemma 8.9, first in Section 8.4 to prove thevarious induction steps, and then again in Section 8.5 to bound the probability thata critical droplet is internally spanned. Lemma 8.25.
Let D ⊂ D (cid:48) be nested S U -droplets.(i) Let (cid:54) β (cid:54) α and (cid:54) β (cid:54) α , and suppose that IH( β , β ) holds. If ξ − (cid:54) w ( D (cid:48) ) − w ( D ) (cid:54) p − β (1 − η ) − η and h ( D (cid:48) ) (cid:54) p − β (1 − η ) − η , then P p (cid:0) ∆( D, D (cid:48) ) (cid:1) (cid:54) p Ω( δ (cid:48) )( w ( D (cid:48) ) − w ( D )) , where δ (cid:48) = δ (cid:48) ( β + β ) , and the constant implicit in Ω( · ) depends only on U .(ii) Suppose that IH( α, α + 1) holds. If ξ − (cid:54) w ( D (cid:48) ) − w ( D ) (cid:54) p − α (1 − η ) − η and h ( D (cid:48) ) (cid:54) ξp α log 1 p , then P p (cid:0) ∆( D, D (cid:48) ) (cid:1) (cid:54) w ( D (cid:48) ) · exp (cid:16) − p O ( ξ ) (cid:0) w ( D (cid:48) ) − w ( D ) (cid:1)(cid:17) , where the implicit constant depends on κ (2 α + 1) . DD (cid:48) S r DD (cid:48) S r Figure 8.
Two possible configurations of the droplets in Lemma 8.25(note that, as in the definition of S r in the proof of the lemma, the u r -sides of S r and D (cid:48) are equal in both cases). The hatching indicatesthe sites that would be assumed to be present for the purposes of theevent that S r is u r -crossed. In both examples it is easy to see thatthe event ∆( D, D (cid:48) ) implies that S r is u r -crossed. NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 63
Proof.
Let S r ⊂ D (cid:48) be the unique maximal u r -strip whose u r -side is equal to thatof D (cid:48) and which does not intersect D ( S r may or may not be a subset of D (cid:48) ,depending on the shapes and relative positions of D and D (cid:48) ; see Figure 8). Define S l similarly on the u l -side of D (cid:48) . We claim that if the event ∆( D, D (cid:48) ) occurs,then S r is u r -crossed and S l is u l -crossed. Indeed, ∆( D, D (cid:48) ) implies that thereexists a strongly connected component L of [ D ∪ ( D (cid:48) ∩ A )] such that D (cid:48) is thesmallest S U -droplet containing L . Now choose x ∈ ∂ ( S r , − u r ), and observe that L ⊂ [ D ∪ ( D (cid:48) ∩ A )] ⊂ [ H u r ( x ) ∪ ( S r ∩ A )], and that L intersects both H u r ( x ) and ∂ ( S r , u r ), since the u r -sides of S r and D (cid:48) are equal. Hence S r is u r -crossed, asclaimed, and similarly S l is u l -crossed.Next, note that h ( S r ) = h ( S l ) = h ( D (cid:48) ), and that we have the easy geometricinequalitiesΩ (cid:0) w ( D (cid:48) ) − w ( D ) (cid:1) = max (cid:8) π ( S r , u r ) , π ( S l , u l ) (cid:9) (cid:54) w ( D (cid:48) ) − w ( D ) , where the implicit constant depends only on U . Since w ( D (cid:48) ) − w ( D ) (cid:62) ξ − , itfollows that max (cid:8) π ( S r , u r ) , π ( S l , u l ) (cid:9) (cid:62) ξ − , since ξ was chosen sufficiently small(depending on U ). Hence, by Lemma 8.18, we have P p (cid:0) ∆( D, D (cid:48) ) (cid:1) (cid:54) p Ω( δ (cid:48) ( β + β ))( w ( D (cid:48) ) − w ( D )) under the assumptions of part (i), and that P p (cid:0) ∆( D, D (cid:48) ) (cid:1) (cid:54) w ( D (cid:48) ) · exp (cid:16) − p O ( ξ ) (cid:0) w ( D (cid:48) ) − w ( D ) (cid:1)(cid:17) under the assumptions of part (ii), as required. (cid:3) Finally, let us prove Lemma 8.19, which bounds the probability of a verticalcrossing. When ¯ α ( u ∗ ) (cid:62) α + 1 (the ‘non-drift’ case), in which case u = u ∗ , the proofis straightforward; indeed, in this case the application of Lemma 8.22 is the sameas in the proof of Lemma 8.18. When α − ( u ∗ ) = ∞ (the ‘drift’ case), and u ∈ S + U issuch that σ ( u ) = p − η , on the other hand, this naive approach no longer works, andthe proof in this case is conceptually a little more difficult, since it requires us to usethe stretched geometry of the u -norm in order to control the unbounded sidewaysgrowth of small sets. This is the only point in the proof of Theorem 8.1 where wespecifically need the u -norm. Proof of Lemma 8.19.
Let 1 (cid:54) β (cid:54) α + 1 and 1 (cid:54) β (cid:54) α , and suppose thatIH( β , β ) holds. Let u ∈ S + U ∩ Q , and let S be a u -crossed u -strip with w ( S ) (cid:54) p − β (1 − η ) − η , h ( S ) (cid:54) p − β (1 − η ) − η , and π ( S, u ) (cid:62) ξ − . (52)We begin with the (easier) ‘non-drift’ case, for which the proof is almost identicalto that of Lemma 8.18. Case 1: u = u ∗ and ¯ α ( u ∗ ) (cid:62) α + 1. Since S is u -crossed by A , there exists a ( u ∗ , α + 1)-partition for S ∩ A , byLemma 8.22. Let S ∪ · · · ∪ S m (cid:48) +1 be a partition of S into u -strips as in Defini-tion 8.21, and note that π ( S, u ∗ ) = h ( S ), and that there are at most O (cid:0) w ( S ) (cid:1) u ∗ -weak ( α + 1)-clusters in each sub-strip S i , where the implicit constant dependson κ = κ ( β + β ). It therefore follows from Lemma 8.24 that S ∩ A admits a( u ∗ , α + 1)-partition with probability at most h ( S ) · max (cid:54) j (cid:54) m (cid:16) − (cid:0) − p α +1 (cid:1) O ( w ( S )) (cid:17) m − j (cid:0) h ( S ) · p α (cid:1) j , (53)where m = (cid:98) h ( S ) / α + 1) κ ( β + β ) (cid:99) . Since w ( S ) (cid:54) p − β (1 − η ) − η and 1 (cid:54) β (cid:54) α + 1, we have 1 − (1 − p α +1 ) O ( w ( S )) = O (cid:0) p α +1 − β (1 − η ) − η (cid:1) (cid:54) p η , Also, since h ( S ) (cid:54) p − β (1 − η ) − η and 1 (cid:54) β (cid:54) α , we have h ( S ) · p α (cid:54) p η . Thus,noting that m (cid:29) h ( S ) (cid:62) ξ − (cid:29) κ ( β + β )), it follows that (53) is at most h ( S ) · max (cid:54) j (cid:54) m p η ( m − j ) · p ηj = h ( S ) · p ηm (cid:54) p δ (cid:48) h ( S ) = p δ (cid:48) π ( S,u ) , as required, where δ (cid:48) = δ (cid:48) ( β + β ) (cid:54) η/ α + 1) κ ( β + β ).We now turn to the ‘drift’ case. Case 2: σ ( u ) = p − η and α − ( u ∗ ) = ∞ .Since S is u -crossed by A , and ¯ α ( u ) = ∞ , by Lemma 8.22 there exists a ( u, λ )-partition for S ∩ A . By Lemma 8.24, this occurs with probability at most π ( S, u ) · max (cid:54) j (cid:54) m (cid:16) − (cid:0) − p λ (cid:1) g u ( S ) (cid:17) m − j (cid:0) π ( S, u ) · p α (cid:1) j , (54)where m := (cid:98) π ( S, u ) / λκ ( β + β ) (cid:99) , and g u ( S ) denotes the number of u -weak λ -clusters in a sub-strip S (cid:48) ⊂ S of u -projection 4 λκ ( β + β ). We claim that g u ( S ) = O (cid:0) w ( S ) · p − λ (1 − η ) (cid:1) , where the implicit constant depends on κ ( β + β ). Indeed, there are O (cid:0) w ( S ) (cid:1) choices for the first site in the u -weak λ -cluster, and at most O (1 /σ ) choices foreach of the remaining λ − − (cid:0) − p λ (cid:1) g u ( S ) (cid:54) O (cid:0) w ( S ) (cid:1) · p − λ (1 − η ) · p λ (cid:54) O (cid:0) w ( S ) (cid:1) · p α (cid:54) p α , since w ( S ) (cid:54) p − β (1 − η ) − η and 1 (cid:54) β (cid:54) α + 1, and recalling from (40) that λ = 5 α/η .Finally, note that π ( S, u ) · p α (cid:54) p η , since π ( S, u ) (cid:54) · h ( S ) (cid:54) · p − β (1 − η ) − η and 1 (cid:54) β (cid:54) α . Thus, noting that m (cid:29) π ( S, u ) (cid:62) ξ − (cid:29) κ ( β + β )), itfollows that (54) is at most π ( S, u ) · max (cid:54) j (cid:54) m p α ( m − j ) · p ηj (cid:54) π ( S, u ) · p ηm (cid:54) p δ (cid:48) π ( S,u ) , where δ (cid:48) = δ (cid:48) ( β + β ) (cid:54) η/ λκ ( β + β ). This completes the proof of the lemma. (cid:3) NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 65
The induction steps.
In this subsection we prove the induction steps. Thefollowing lemma will be used to deduce the implications IH( β, β ) ⇒ IH( β + 1 , β )and IH( β + 1 , β ) ∧ IH( β, β + 1) ⇒ IH( β + 1 , β + 1).For convenience, we shall occasionally use the notation exp p ( x ) := p x . Lemma 8.26.
Let (cid:54) β (cid:54) α and (cid:54) β (cid:54) α with β (cid:54) β + 1 , and suppose that IH( β , β ) holds. Let D be an S U -droplet such that p − β (1 − η ) − η (cid:54) w ( D ) (cid:54) p − ( β +1)(1 − η ) − η and h ( D ) (cid:54) p − β (1 − η ) − η . Then P p (cid:0) I × ( D ) (cid:1) (cid:54) p Ω( δ (cid:48) ) w ( D ) , where δ (cid:48) = δ (cid:48) ( β + β ) , and the implicit constant depends only on U .Proof. We shall use the hierarchies framework from Section 8.1 with β = β . Tobegin, recall the bound from Lemma 8.9: P p (cid:0) I × ( D ) (cid:1) (cid:54) (cid:88) H∈H D (cid:18) (cid:89) u ∈ L ( H ) P p (cid:0) I × ( D u ) (cid:1)(cid:19)(cid:18) (cid:89) u → v P p (cid:0) ∆( D v , D u ) (cid:1)(cid:19) . (55)In order to use this bound, we need estimates for the probability that a seed isinternally spanned, the probability of the event ∆( D v , D u ), and the number of goodhierarchies for D .First, for each u ∈ L ( H ) we have w ( D u ) (cid:54) p − β (1 − η ) − η , by Definition 8.5. Also,since D u ⊂ D , we have h ( D ) (cid:54) p − β (1 − η ) − η . Hence, by IH( β , β ), P p (cid:0) I × ( D u ) (cid:1) (cid:54) p δw ( D u ) , (56)where δ = δ ( β + β ).We bound the probability of the event ∆( D v , D u ) using Lemma 8.25. Recall that w ( D u ) − w ( D v ) (cid:54) p − β (1 − η ) − η , by Definition 8.5, and note that h ( D v ) (cid:54) h ( D u ) (cid:54) p − β (1 − η ) − η , since D u ⊂ D . However, in order to use Lemma 8.25, we also needthe lower bound w ( D u ) − w ( D v ) (cid:62) ξ − , which Definition 8.5 does not guarantee.Therefore, we can deduce from Lemma 8.25 that P p (cid:0) ∆( D v , D u ) (cid:1) (cid:54) p Ω( δ (cid:48) )( w ( D u ) − w ( D v )) , (57)where δ (cid:48) = δ (cid:48) ( β + β ), and the constant implicit in Ω( · ) depends only on U , only forthose pairs { u, v } ⊂ V ( G H ) with N → G H ( u ) = { v } such that w ( D u ) − w ( D v ) (cid:62) ξ − .We now divide into two cases according to the number of big seeds of H (i.e., thenumber of leaves u ∈ L ( H ) such that w ( D u ) (cid:62) p − β (1 − η ) − η /
3; see Definition 8.11).The idea is as follows: if there are ‘few’ big seeds, then the number of hierarchiesis small enough (by Lemma 8.13) that we can uniformly bound the probability ofeach; on the other hand, if there are ‘many’ big seeds, then the contribution to (55)from the big seeds alone outweighs the combinatorial cost of counting the goodhierarchies. Thus, set B := p − η and let H (1) := (cid:8) H ∈ H D : b ( H ) (cid:54) B (cid:9) and H (2) := H D \ H (1) . Bounding the sum over
H ∈ H (2) is the simpler case. Indeed, observe that (cid:88)
H∈H (2) (cid:89) u ∈ L ( H ) P p (cid:0) I × ( D u ) (cid:1) (cid:54) (cid:88) b>B |H bD | · exp p (cid:18) δ · b · p − β (1 − η ) − η (cid:19) , using the notation exp p ( x ) := p x , and using (56) and the definition of a big seed.Moreover, by Lemma 8.13, for each b we have (cid:12)(cid:12) H bD (cid:12)(cid:12) (cid:54) exp p (cid:16) − O (cid:16) b · w ( D ) · p β (1 − η )+ η (cid:1)(cid:17) (cid:54) exp p (cid:16) − O (cid:0) b · p − η (cid:17)(cid:17) , since w ( D ) (cid:54) p − ( β +1)(1 − η ) − η . Hence, since β (cid:62) (cid:88) H∈H (2) (cid:89) u ∈ L ( H ) P p (cid:0) I × ( D u ) (cid:1) (cid:54) (cid:88) b>B exp p (cid:18) δ · b · p − β (1 − η ) − η (cid:19) (cid:54) p δw ( D ) / , (58)where the last step follows since B = p − η and w ( D ) (cid:54) p − ( β +1)(1 − η ) − η .To deal with H (1) , first we use the two estimates (56) and (57) to obtain (cid:88) H∈H (1) (cid:18) (cid:89) u ∈ L ( H ) P p (cid:0) I × ( D u ) (cid:1)(cid:19)(cid:18) (cid:89) u → v P p (cid:0) ∆( D v , D u ) (cid:1)(cid:19) (cid:54) (cid:88) H∈H (1) exp p (cid:32) δ (cid:88) u ∈ L ( H ) w ( D u ) + Ω( δ (cid:48) ) (cid:18) (cid:88) u → v (cid:16) w ( D u ) − w ( D v ) (cid:17) − ξ − | V ( G H ) | (cid:19)(cid:33) , (59)where the final term in the exponential accounts for the fact that we can only use (57)when w ( D v ) − w ( D u ) (cid:62) ξ − . Now, by Lemma 8.10, we have (cid:88) u ∈ L ( H ) w ( D u ) + (cid:88) u → v (cid:16) w ( D u ) − w ( D v ) (cid:17) (cid:62) w ( D ) − O (cid:0) | V ( G H ) | (cid:1) for every H ∈ H (1) , and by (34), we have (cid:12)(cid:12) V ( G H ) (cid:12)(cid:12) = O (cid:0) B · w ( D ) · p β (1 − η )+ η (cid:1) = o (cid:0) w ( D ) (cid:1) . Thus, recalling that δ ( β + β ) (cid:62) δ (cid:48) ( β + β ), the right-hand side of (59) is at most (cid:88) H∈H (1) p Ω( δ (cid:48) ) w ( D ) (cid:54) (cid:88) b (cid:54) B (cid:12)(cid:12) H bD (cid:12)(cid:12) · p Ω( δ (cid:48) ) w ( D ) . (60)Now, by Lemma 8.13, we have (cid:12)(cid:12) H bD (cid:12)(cid:12) (cid:54) exp p (cid:16) − O (cid:0) b · w ( D ) · p β (1 − η )+ η (cid:1)(cid:17) (cid:54) e w ( D ) , since b (cid:54) B = p − η and β (cid:62)
1, so the right-hand side of (60) is at most B · e w ( D ) · p Ω( δ (cid:48) ) w ( D ) (cid:54) p Ω( δ (cid:48) ) w ( D ) , since B = p − η (cid:54) e w ( D ) . Combining this bound with (55) and (58), and recallingthat δ ( β + β ) (cid:62) δ (cid:48) ( β + β ), it follows that P p (cid:0) I × ( D ) (cid:1) (cid:54) p Ω( δ (cid:48) ) w ( D ) , as required. (cid:3) NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 67
Before we can prove a corresponding lemma for ‘tall’ droplets, we need one moretechnical lemma, which says that if a droplet D is ‘crossed’ with help from both the u ∗ -side and the ( − u ∗ )-side, then, in a certain sense (which is made explicit in thelemma), the droplet is at least ‘half crossed’ with help from just one side. This willallow us to transfer from a droplet that is ‘crossed’ with help from both sides to a u -crossed u -strip, for an appropriate u . One such application of the lemma is shownin Figure 9, although note that some of the labelling is different, to reflect the setupin Lemma 8.28. D i − D i +1 D i T i L i x i H u ( x i ) − u ∗ u ∗ u r u l Figure 9.
The figure depicts the application of Lemma 8.27 in theproof of Lemma 8.28 assuming α − ( u ∗ ) = ∞ . T i is the minimal u -stripsuch that L i ⊂ T i ∪ H u ( x i ), where σ ( u ) = p − η , and is u -crossed by A .Let us say that an ordered partition D = D ∪ · · · ∪ D t of an S U -droplet D is a vertical partition of D if each D i is an S U -droplet of the form D ∩ H u ∗ ( a ) ∩ H − u ∗ ( b )for some a, b ∈ R , and D i lies between D i − and D i +1 for every 2 (cid:54) i (cid:54) t − Lemma 8.27.
Let D (cid:48) = D u ∗ ∪ D ∪ D − u ∗ be a vertical partition of an S U -droplet D (cid:48) ,and suppose that Z := (cid:2) D u ∗ ∪ ( D ∩ A ) ∪ D − u ∗ (cid:3) (61) contains a strongly connected component Z (cid:48) such that D u ∗ ∪ D − u ∗ ⊂ Z (cid:48) . Then thereexists a set L ⊂ D with h ( L ) (cid:62) h ( D ) / − κ such that, for some u ∈ { u ∗ , − u ∗ } , L ∪ D u is a strongly connected component of (cid:2) D u ∪ ( D ∩ A ) (cid:3) .Proof. For each u ∈ { u ∗ , − u ∗ } , let Z u be the strongly connected component of (cid:2) D u ∪ ( D ∩ A ) (cid:3) containing D u , and note that Z u ⊂ D (cid:48) , since D (cid:48) is an S U -droplet. If the set Z u ∗ ∪ Z − u ∗ is strongly connected, then set L u := Z u ∩ D for each u ∈{ u ∗ , − u ∗ } , and observe that h ( L u ∗ ) + h ( L − u ∗ ) (cid:62) h ( D ) − κ, as required.So suppose that Z u ∗ ∪ Z − u ∗ is not strongly connected, and let Y be the collectionof strongly connected components of (cid:2) ( D ∩ A ) \ ( Z u ∗ ∪ Z − u ∗ ) (cid:3) . Then Z u ∪ Y is notstrongly connected for any u ∈ { u ∗ , − u ∗ } and Y ∈ Y , and thus Y ∪ (cid:8) Z u ∗ , Z − u ∗ (cid:9) is precisely the collection of strongly connected components of Z . But Z containsa strongly connected component containing both D u ∗ and D − u ∗ , and so this is acontradiction, which completes the proof of the lemma. (cid:3) The next lemma will be used to deduce the implications IH( β, β ) ⇒ IH( β, β + 1)and IH( β + 1 , β ) ∧ IH( β, β + 1) ⇒ IH( β + 1 , β + 1), and also to bound the probabilitythat a critical droplet is internally spanned. It follows by combining Lemma 8.27with our bound on the probability of a vertical crossing, Lemma 8.19. Lemma 8.28.
Let (cid:54) β (cid:54) α and β (cid:54) β (cid:54) β + 1 , and suppose that IH( β , β ) holds. Let D be an S U -droplet such that w ( D ) (cid:54) p − β (1 − η ) − η and h ( D ) (cid:62) max (cid:8) p − η w ( D ) , p − η (cid:9) . Then P p (cid:0) I × ( D ) (cid:1) (cid:54) p δ (cid:48) h ( D ) / , where δ (cid:48) = δ (cid:48) ( β + β ) .Proof. Let D = D ∪ · · · ∪ D m +1 be a vertical partition of D such that h ( D ) = h ( D m +1 ) = 0, and12 · max (cid:8) p − η w ( D ) , ξ − (cid:9) (cid:54) h ( D i ) (cid:54) · max (cid:8) p − η w ( D ) , ξ − (cid:9) for each 1 (cid:54) i (cid:54) m ; this is possible by the lower bound on h ( D ). Let us also define,for each 1 (cid:54) i (cid:54) m , droplets D ( i ) u ∗ := D ∪ · · · D i − and D ( i ) − u ∗ := D i +1 ∪ · · · D m +1 ,and note that D = D ( i ) u ∗ ∪ D i ∪ D ( i ) − u ∗ is a vertical partition of D .Now, if D is internally spanned, then for each 1 (cid:54) i (cid:54) m the set Z := (cid:2) D ( i ) u ∗ ∪ ( D i ∩ A ) ∪ D ( i ) − u ∗ (cid:3) contains a strongly connected component Z (cid:48) such that D ( i ) u ∗ ∪ D ( i ) − u ∗ ⊂ Z (cid:48) . ByLemma 8.27, it follows that there exists L ⊂ D i with h ( L ) (cid:62) h ( D i ) / − κ suchthat L ∪ D ( i ) u is a strongly connected component of (cid:2) D ( i ) u ∪ ( D i ∩ A ) (cid:3) for some u ∈ { u ∗ , − u ∗ } . Let E i denote the event that such a set L exists in D i . NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 69
Claim 8.29. P p ( E i ) (cid:54) p δ (cid:48) h ( D i ) / for each (cid:54) i (cid:54) m , where δ (cid:48) = δ (cid:48) ( β + β ) .Proof of Claim 8.29. Fix 1 (cid:54) i (cid:54) m , and suppose that E i occurs. Without loss ofgenerality, let L i ⊂ D i be such that h ( L i ) (cid:62) h ( D i ) / − κ , and L i ∪ D i − is a stronglyconnected component of (cid:2) D i − ∪ ( D i ∩ A ) (cid:3) . Note that each site of D i ∩ A used toinfect a member of L i is itself a member of L i (since L i ∪ D i − is a component, and κ > ν ), so L i ∪ D i − is also a strongly connected component of (cid:2) D i − ∪ ( L i ∩ A ) (cid:3) .We will divide into two cases according to whether or not u ∗ is a drift direction. Case 1: ¯ α ( u ∗ ) (cid:62) α + 1.Observe that the minimal u ∗ -strip S i containing L i is u ∗ -crossed. Note also that w ( S i ) = w ( L i ) (cid:54) w ( D ) (cid:54) p − β (1 − η ) − η , and that h ( S i ) (cid:54) h ( D i ) (cid:54) · max (cid:8) p − η w ( D ) , ξ − (cid:9) (cid:54) p − β (1 − η ) − η , since β (cid:54) β + 1. Note also that π ( S i , u ∗ ) = h ( S i ) = h ( L i ) (cid:62) h ( D i ) / (cid:62) ξ − , since h ( D i ) (cid:62) ξ − (cid:29) κ . Therefore, S i satisfies the conditions of Lemma 8.19, andis thus u ∗ -crossed with probability at most p δ (cid:48) h ( S i ) (cid:54) p δ (cid:48) h ( D i ) / .Now, recall that S i is a T -droplet, where T = { u ∗ , − u ∗ , u ⊥ , − u ⊥ } , and so is definedby four half-planes, one for each element of T . There are at most w ( D ) · h ( D i ) choicesfor each, and hence at most p − α choices for S i . Thus, taking a union bound over u ∗ -strips S i , and recalling that h ( D i ) (cid:62) ξ − (cid:29) δ (cid:48) ( β + β ) − , it follows that P p ( E i ) (cid:54) p − α · p δ (cid:48) h ( D i ) / (cid:54) p δ (cid:48) h ( D i ) / , as claimed. Case 2: α − ( u ∗ ) = ∞ .Let x i be the element of R at the intersection of the u l and ( − u ∗ )-sides of D i , let u ∈ S + U ∩ Q be such that σ ( u ) = p − η , and let T i be the minimal u -strip such that L i ⊂ H u ( x i ) ∪ T i (see Figure 9). Observe that T i is u -crossed by D i ∩ A ; indeed, since D i − ⊂ H u ( x i )and L i ⊂ (cid:2) D i − ∪ ( L i ∩ A ) (cid:3) , it follows that L i ⊂ [ H u ( x i ) ∪ ( T i ∩ A )].We next claim that w ( T i ) and h ( T i ) satisfy the conditions of Lemma 8.19. Indeed,we have w ( T i ) = w ( L i ) (cid:54) w ( D ) (cid:54) p − β (1 − η ) − η , and h ( T i ) (cid:54) h ( D i ) + σ ( u ) · w ( D ) (cid:54) p − β (1 − η ) − η , since β (cid:54) β + 1 (cf. Case 1). Moreover, we have π ( T i , u ) (cid:62) h ( T i ) − · σ ( u ) · w ( D ) (cid:62) h ( T i )2 (cid:62) h ( L i )2 (cid:62) ξ, Indeed, L i ⊂ [ H u ∗ ( x ) ∪ ( S i ∩ A )] for any x ∈ ∂ ( S i , − u ∗ ), since D i − ⊂ H u ∗ ( x ) and L i is astrongly connected component of (cid:2) D i − ∪ ( L i ∩ A ) (cid:3) . since σ ( u ) = p − η and h ( T i ) (cid:62) h ( L i ) (cid:62) h ( D i ) / (cid:62) · max (cid:8) p − η w ( D ) , ξ − (cid:9) . Itfollows, by Lemma 8.19, that T i is u -crossed with probability at most p δ (cid:48) h ( T i ) (cid:54) p δ (cid:48) h ( D i ) / .Now, as in Case 1, there are at most p − α choices for T i . Thus, taking a unionbound over u -strips T i , and recalling again that h ( D i ) (cid:62) ξ − (cid:29) δ (cid:48) ( β + β ) − , itfollows that P p ( E i ) (cid:54) p − α · p δ (cid:48) h ( D i ) / (cid:54) p δ (cid:48) h ( D i ) / , as claimed. (cid:3) Finally, note that the events E , . . . , E m are independent, since E i depends onlyon the set D i ∩ A . Therefore, by Claim 8.29, P p (cid:0) I × ( D ) (cid:1) (cid:54) exp p (cid:18) δ (cid:48) m (cid:88) i =1 h ( D i ) (cid:19) (cid:54) p δ (cid:48) h ( D ) / , and this completes the proof of the lemma. (cid:3) We are now ready to prove Lemma 8.3.
Proof of Lemma 8.3.
We shall prove by induction on β + β that IH( β , β ) holdsfor every pair ( β , β ) ∈ N with2 (cid:54) β + β (cid:54) α + 1 and | β − β | (cid:54) . Observe first that IH(1 ,
1) follows from Lemma 6.18, since δ (2) was chosen (in (31))to be sufficiently small (depending on η ). The induction steps are of three differenttypes, which are dealt with in the following three claims. The first follows fromLemma 8.26, the second from Lemma 8.28, and the third requires both. Claim 8.30.
For each (cid:54) β (cid:54) α we have IH( β, β ) ⇒ IH( β + 1 , β ) . Proof.
Let D be a droplet with w ( D ) (cid:54) p − ( β +1)(1 − η ) − η and h ( D ) (cid:54) p − β (1 − η ) − η . We are required to show that P p (cid:0) I × ( D ) (cid:1) (cid:54) p δ max { w ( D ) ,h ( D ) } , where δ = δ (2 β + 1).If w ( D ) (cid:54) p − β (1 − η ) − η then this follows immediately from IH( β, β ) (since we chose δ (2 β + 1) (cid:54) δ (2 β ) in (31)), so we may assume that w ( D ) (cid:62) p − β (1 − η ) − η . Now,applying Lemma 8.26 with β = β = β , it follows that P p (cid:0) I × ( D ) (cid:1) (cid:54) p Ω( δ (cid:48) (2 β )) w ( D ) (cid:54) p δ (2 β +1) w ( D ) , as required, since we chose δ (2 β + 1) (cid:28) δ (cid:48) (2 β ) in (38). (cid:3) NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 71
Claim 8.31.
For each (cid:54) β (cid:54) α we have IH( β, β ) ⇒ IH( β, β + 1) . Proof.
Let D be a droplet with w ( D ) (cid:54) p − β (1 − η ) − η and h ( D ) (cid:54) p − ( β +1)(1 − η ) − η . We again need to show that P p (cid:0) I × ( D ) (cid:1) (cid:54) p δ max { w ( D ) ,h ( D ) } , where δ = δ (2 β + 1).Note that if h ( D ) (cid:54) p − β (1 − η ) − η then this follows immediately from IH( β, β ), asbefore, so we may assume that h ( D ) (cid:62) p − β (1 − η ) − η , which implies that h ( D ) (cid:62) max (cid:8) p − η w ( D ) , p − η (cid:9) . Hence, applying Lemma 8.28 with β = β = β , we obtain P p (cid:0) I × ( D ) (cid:1) (cid:54) p δ (cid:48) (2 β ) h ( D ) / (cid:54) p δ (2 β +1) h ( D ) , as required, since we chose δ (2 β + 1) (cid:28) δ (cid:48) (2 β ) in (38). (cid:3) Claim 8.32.
For each (cid:54) β (cid:54) α − we have (cid:0) IH( β + 1 , β ) ∧ IH( β, β + 1) (cid:1) ⇒ IH( β + 1 , β + 1) . Proof.
Let D be an S U -droplet with w ( D ) (cid:54) p − ( β +1)(1 − η ) − η and h ( D ) (cid:54) p − ( β +1)(1 − η ) − η . This time we are required to show that P p (cid:0) I × ( D ) (cid:1) (cid:54) p δ (2 β +2) max { w ( D ) ,h ( D ) } . Notethat, since we chose δ (2 β + 2) (cid:54) δ (2 β + 1) in (31), this follows immediately fromIH( β, β + 1) if w ( D ) (cid:54) p − β (1 − η ) − η , and from IH( β + 1 , β ) if h ( D ) (cid:54) p − β (1 − η ) − η .We may therefore assume thatmin (cid:8) w ( D ) , h ( D ) (cid:9) (cid:62) p − β (1 − η ) − η . Suppose first that w ( D ) (cid:62) h ( D ). Then, applying Lemma 8.26 with β = β and β = β + 1, it follows that P p (cid:0) I × ( D ) (cid:1) (cid:54) p Ω( δ (cid:48) (2 β +1)) w ( D ) (cid:54) p δ (2 β +2) w ( D ) , as required, since we chose δ (2 β + 2) (cid:28) δ (cid:48) (2 β + 1) in (38).On the other hand, if w ( D ) (cid:54) h ( D ) then we have h ( D ) (cid:62) max (cid:8) p − η w ( D ) , p − η (cid:9) . Hence, applying Lemma 8.28 with β = β + 1 and β = β , we obtain P p (cid:0) I × ( D ) (cid:1) (cid:54) p δ (cid:48) (2 β ) h ( D ) / (cid:54) p δ (2 β +2) h ( D ) , as required, since we chose δ (2 β + 2) (cid:28) δ (cid:48) (2 β + 1) in (38). (cid:3) Together with IH(1 , α + 1 , α ) and IH( α, α + 1), whichcompletes the proof of the lemma. (cid:3) Internally spanned critical droplets.
Recall from Definition 2.5 that wecall an S U -droplet D critical if one of the following holds:( T ) w ( D ) (cid:54) p − α − / and ξp α log p (cid:54) h ( D ) (cid:54) ξp α log p , or( L ) p − α − / (cid:54) w ( D ) (cid:54) p − α − / and h ( D ) (cid:54) ξp α log p ,where ξ > δ > δ (cid:28) ξ (cid:28) δ (cid:48) (2 α + 1). Lemma 8.33. If D is a critical droplet then P p (cid:0) I × ( D ) (cid:1) (cid:54) exp (cid:32) − δp α (cid:18) log 1 p (cid:19) (cid:33) . To prove the lemma for ‘tall’ droplets (type ( T )), we simply apply Lemmas 8.3and 8.28. For ‘long’ droplets (type ( L )), on the other hand, we need to apply themethod of hierarchies, as in the proof of Lemma 8.26, together with Lemmas 8.3and 8.25, which we use to bound the probabilities of the events in Definition 8.6. Proof of Lemma 8.33.
We will prove that the lemma holds with δ = ξ · δ (cid:48) (2 α + 1)8 . Let D be a critical droplet, and suppose first that D is of type ( T ). Then, since w ( D ) (cid:54) p − α − / and h ( D ) (cid:62) ξp α log p , and recalling that η = (10 α ) − , we have w ( D ) (cid:54) p − ( α +1)(1 − η ) − η and h ( D ) (cid:62) max (cid:8) p − η w ( D ) , p − η (cid:9) . We may therefore apply Lemma 8.28 with β = α + 1 and β = α , since IH( α + 1 , α )holds by Lemma 8.3. This gives P p (cid:0) I × ( D ) (cid:1) (cid:54) p δ (cid:48) (2 α +1) h ( D ) / (cid:54) exp (cid:32) − δp α (cid:18) log 1 p (cid:19) (cid:33) , as required.So suppose from now on that D is of type ( L ); in this case we will prove thefollowing much stronger bound: P p (cid:0) I × ( D ) (cid:1) (cid:54) exp (cid:16) − p − α − / (cid:17) . (62)We apply the hierarchies framework, as in the proof of Lemma 8.26, but with β = α .By Lemma 8.9, we have P p (cid:0) I × ( D ) (cid:1) (cid:54) (cid:88) H∈H D (cid:18) (cid:89) u ∈ L ( H ) P p (cid:0) I × ( D u ) (cid:1)(cid:19)(cid:18) (cid:89) u → v P p (cid:0) ∆( D v , D u ) (cid:1)(cid:19) . (63) NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 73
Now, if u ∈ L ( H ), then since D is of type ( L ), and by Definitions 8.4 and 8.5, w ( D u ) (cid:54) p − α (1 − η ) − η and h ( D u ) (cid:54) h ( D ) (cid:54) ξp α log 1 p (cid:54) p − ( α +1)(1 − η ) − η . Since IH( α, α + 1) holds, by Lemma 8.3, it follows that P p (cid:0) I × ( D u ) (cid:1) (cid:54) p δ (2 α +1) w ( D u ) , (64)for every u ∈ L ( H ).Next, if N → G H ( u ) = { v } , then by Definitions 8.4 and 8.5 we have w ( D u ) − w ( D v ) (cid:54) p − α (1 − η ) − η and h ( D v ) (cid:54) h ( D u ) (cid:54) h ( D ) (cid:54) ξp α log 1 p , as above. Since IH( α, α + 1) holds, by Lemma 8.3, it follows by Lemma 8.25 that if w ( D u ) − w ( D v ) (cid:62) ξ − , then P p (cid:0) ∆( D v , D u ) (cid:1) (cid:54) exp (cid:16) − p O ( ξ ) (cid:0) w ( D u ) − w ( D v ) (cid:1)(cid:17) , (65)where the implicit constant depends on κ (2 α + 1).As in Lemma 8.26, we divide into two cases according to whether H has many orfew big seeds. Thus, let B := p − / and let H (1) := (cid:8) H ∈ H D : b ( H ) (cid:54) B (cid:9) and H (2) := H D \ H (1) . Bounding the sum in (63) over
H ∈ H (2) is again the easier case. Indeed, by (64), (cid:88)
H∈H (2) (cid:89) u ∈ L ( H ) P p (cid:0) I × ( D u ) (cid:1) (cid:54) (cid:88) b>B |H bD | · exp p (cid:18) δ (2 α + 1) · b · p − α (1 − η ) − η (cid:19) , (66)and by Lemma 8.13, for each b ∈ N we have (cid:12)(cid:12) H bD (cid:12)(cid:12) (cid:54) exp p (cid:16) − O (cid:16) b · w ( D ) · p α (1 − η )+ η (cid:1)(cid:17) (cid:54) exp p (cid:16) − O (cid:0) b · p − η (cid:17)(cid:17) , since w ( D ) (cid:54) p − ( α +1)(1 − η ) − η . The right-hand side of (66) is therefore at most (cid:88) b>B exp p (cid:18) Ω (cid:0) δ (2 α + 1) (cid:1) · b · p − α (1 − η ) − η (cid:19) (cid:54) exp (cid:16) − p − α − / (cid:17) , (67)since η = (10 α ) − and B = p − / .For hierarchies with few big seeds, observe that by (64) and (65), we have (cid:89) u ∈ L ( H ) P p (cid:0) I × ( D u ) (cid:1) (cid:89) u → v P p (cid:0) ∆( D v , D u ) (cid:1) (cid:54) exp (cid:20) − p O ( ξ ) (cid:18) (cid:88) u ∈ L ( H ) w ( D u ) + (cid:88) u → v (cid:16) w ( D u ) − w ( D v ) (cid:17) − ξ − | V ( G H ) | (cid:19)(cid:21) (68) for each H ∈ H (1) , since δ (2 α + 1) log(1 /p ) > p O ( ξ ) , and where the final term inthe exponential takes account of the condition w ( D u ) − w ( D v ) (cid:62) ξ − , which wasassumed in the proof of (65). Now, recall that (cid:88) u ∈ L ( H ) w ( D u ) + (cid:88) u → v (cid:16) w ( D u ) − w ( D v ) (cid:17) (cid:62) w ( D ) − O (cid:0) | V ( G H ) | (cid:1) by Lemma 8.10, (cid:12)(cid:12) V ( G H ) (cid:12)(cid:12) = O (cid:0) B · w ( D ) · p α (1 − η )+ η (cid:1) = o (cid:0) w ( D ) (cid:1) , by (34), and (cid:12)(cid:12) H bD (cid:12)(cid:12) (cid:54) exp p (cid:16) − O (cid:0) b · w ( D ) · p α (1 − η )+ η (cid:1)(cid:17) (cid:54) exp (cid:16) p / w ( D ) (cid:17) , by Lemma 8.13, since α (cid:62) η = (10 α ) − and b (cid:54) B = p − / . Hence, summing (68)over H ∈ H (1) , and using (63) and the bound (67) proved above for
H ∈ H (1) , itfollows that P p (cid:0) I × ( D ) (cid:1) (cid:54) B (cid:88) b =1 (cid:12)(cid:12) H bD (cid:12)(cid:12) exp (cid:16) − p O ( ξ ) w ( D ) (cid:17) + exp (cid:16) − p − α − / (cid:17) , (cid:54) B exp (cid:16) − p O ( ξ ) w ( D ) (cid:17) (cid:54) exp (cid:16) − p − α − / (cid:17) , since w ( D ) (cid:62) p − α − / and ξ > (cid:3) The proof of Theorem 8.1.
We need one final lemma in order to deduceTheorem 8.1 from Lemma 8.33. It is a simple consequence of Lemma 6.16.
Lemma 8.34. If n (cid:62) p − α and [ A ] = Z n , then there exists a critical droplet that isinternally spanned by A .Proof. Run the spanning algorithm on Z n , with initial set A . Since [ A ] = Z n , wewill (at some point in the algorithm) obtain an internally spanned droplet D withmax { w ( D ) , h ( D ) } (cid:62) p − α . Let D be the first such droplet to appear in thealgorithm, and suppose first that w ( D ) (cid:54) p − α − / , so h ( D ) (cid:62) p − α . Since D is internally spanned, by applying Lemma 6.16 with u = u ∗ and k = ξp α log p , weobtain an internally spanned droplet D ⊂ D with ξp α log 1 p (cid:54) h ( D ) (cid:54) ξp α log 1 p , (69)so D is a type ( T ) critical droplet.On the other hand, if w ( D ) (cid:62) p − α − / then we can apply Lemma 6.16 with u = u ⊥ and k = p − α − / to obtain an internally spanned droplet D ⊂ D with p − α − / (cid:54) w ( D ) (cid:54) p − α − / . (70)If h ( D ) (cid:54) ξp α log p then D is a type ( L ) critical droplet, in which case we are done,so assume not. Now, applying Lemma 6.16 with u = u ∗ and k = ξp α log p , we obtain NIVERSALITY OF TWO-DIMENSIONAL CRITICAL CELLULAR AUTOMATA 75 an internally spanned droplet D ⊂ D such that (69) holds. Since w ( D ) (cid:54) w ( D ) (cid:54) p − α − / , by (70), it follows that D is a type ( T ) critical droplet, as required. (cid:3) We now have all the tools we need to complete the proof of Theorem 8.1, andhence Theorem 1.4.
Proof of Theorem 8.1.
Let ε > ε (cid:28) δ . Set p = (cid:18) ε (log log n ) log n (cid:19) /α , and let A be a p -random subset of Z n . We claim that P p (cid:0) [ A ] = Z n (cid:1) → n → ∞ .Indeed, if [ A ] = Z n then, by Lemma 8.34, there exists a critical droplet D thatis internally spanned by A . By Lemma 8.33, the probability that D is internallyspanned is at most exp (cid:32) − δp α (cid:18) log 1 p (cid:19) (cid:33) , and there are at most n p − α (cid:54) n critical droplets in Z n . Hence P p (cid:0) [ A ] = Z n (cid:1) (cid:54) n exp (cid:32) − δp α (cid:18) log 1 p (cid:19) (cid:33) → n → ∞ , as required, since ε (cid:28) δ . This completes the proof of the theorem. (cid:3) Conjectures for higher dimensions
We conclude by briefly discussing the U -bootstrap percolation models in higherdimensions. Fix an integer d (cid:62) U be a d -dimensional update family.The definition of the stable set S = S ( U ) is the natural generalization of the two-dimensional definition: S := (cid:8) u ∈ S d − : [ H du ] = H du (cid:9) , where H du := (cid:8) x ∈ Z d : (cid:104) x, u (cid:105) < (cid:9) is the discrete half-space in Z d with normal u ∈ S d − . Observe that, as in twodimensions, it is easy to show that the dichotomy [ H du ] ∈ (cid:8) H du , Z d (cid:9) holds for anyunit vector u ∈ S d − .Given a set T ⊂ S d − , we write int( T ) for the interior of T in the usual topologyon S d − (induced by geodesic distance). Generalizing Definition 1.1, we classify d -dimensional update familes as follows. Definition 9.1. A d -dimensional update family is: • subcritical if int( C ∩ S ) (cid:54) = ∅ for every hemisphere C ⊂ S d − ; • critical if there exists a hemisphere C ⊂ S d − such that int( C ∩ S ) = ∅ and if C ∩ S (cid:54) = ∅ for every open hemisphere C ⊂ S d − ; • supercritical if C ∩ S = ∅ for some open hemisphere C ⊂ S .As in two dimensions, the subcritical/critical/supercritical trichotomy dependsonly on the stable set S . However, we expect there to be a further subdivision ofcritical families into d − r for which the modelbehaves (broadly) like the classical r -neighbour model. Conjecture 9.2.
Let U be a d -dimensional bootstrap percolation update family. ( i ) If U is subcritical then p c ( Z d , U ) > . ( ii ) If U is critical then there exist r ∈ { , . . . , d } and α ∈ Q such that p c ( Z dn , U ) = (cid:32) ( r − n (cid:33) α + o (1) . ( iii ) If U is supercritical then p c ( Z dn , U ) = n − Θ(1) . The conjecture for supercritical families is likely to be relatively straightforward,since the lower bound is trivial, the main challenge being to find the correct gen-eralization of quasi-stability to higher dimensions. The conjecture for subcriticalfamilies was originally made by Balister, Bollob´as, Przykucki and Smith [2]. Forcritical families, one might hope to prove an even sharper result, along the lines ofTheorem 1.4, but even the much weaker bounds conjectured above appear to be farout of reach with current techniques.
Acknowledgement
This work was started during a visit of the second author to IMPA in the springof 2013. He would like to thank IMPA for its hospitality. The authors would alsolike to express their sincere gratitude to the anonymous referee, who read this longpaper extremely carefully, pointed out several small gaps in the original proof, andmade a large number of additional helpful comments on the paper.
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