Subcritical U -bootstrap percolation models have non-trivial phase transitions
aa r X i v : . [ m a t h . P R ] S e p SUBCRITICAL U -BOOTSTRAP PERCOLATION MODELS HAVENON-TRIVIAL PHASE TRANSITIONS PAUL BALISTER, B´ELA BOLLOB ´AS, MICHA L PRZYKUCKI, AND PAUL SMITH
Abstract.
We prove that there exist natural generalizations of the classical bootstrappercolation model on Z that have non-trivial critical probabilities, and moreover wecharacterize all homogeneous, local, monotone models with this property.Van Enter [28] (in the case d = r = 2) and Schonmann [25] (for all d > r >
2) provedthat r -neighbour bootstrap percolation models have trivial critical probabilities on Z d forevery choice of the parameters d > r >
2: that is, an initial set of density p almost surelypercolates Z d for every p >
0. These results effectively ended the study of bootstrappercolation on infinite lattices.Recently Bollob´as, Smith and Uzzell [8] introduced a broad class of percolation mod-els called U -bootstrap percolation, which includes r -neighbour bootstrap percolationas a special case. They divided two-dimensional U -bootstrap percolation models intothree classes – subcritical, critical and supercritical – and they proved that, like classi-cal 2-neighbour bootstrap percolation, critical and supercritical U -bootstrap percolationmodels have trivial critical probabilities on Z . They left open the question as to whathappens in the case of subcritical families. In this paper we answer that question: weshow that every subcritical U -bootstrap percolation model has a non-trivial critical prob-ability on Z . This is new except for a certain ‘degenerate’ subclass of symmetric modelsthat can be coupled from below with oriented site percolation. Our results re-open thestudy of critical probabilities in bootstrap percolation on infinite lattices, and they allowone to ask many questions of subcritical bootstrap percolation models that are typicallyasked of site or bond percolation. Introduction
Bootstrap percolation on infinite lattices.
The classical r -neighbour bootstrappercolation model was introduced by Chalupa, Leath and Reich [12] in order to modelcertain physical interacting particle systems. Given a graph G = ( V, E ), usually taken tobe Z d or [ n ] d , a subset A ⊂ V of the set of vertices of G is chosen by including verticesindependently at random with probability p . We write A ∼ Bin(
V, p ) to denote that theset A has this distribution and P p for the product probability measure. The vertices in A are said to be infected . Set A = A and then, for t = 0 , , , . . . , let A t +1 = A t ∪ (cid:8) v ∈ V : | N ( v ) ∩ A t | > r (cid:9) , where N ( v ) is the set of neighbours of v in G . Thus, infected vertices remain infectedforever, and uninfected vertices become infected when at least r of their neighbours in G Mathematics Subject Classification.
Key words and phrases.
Bootstrap percolation, phase transitions.The second author is partially supported by NSF grant DMS 1301614 and MULTIPLEX no. 317532.The third author is supported by MULTIPLEX no. 317532. The fourth author is supported by a CNPqbolsa PDJ. are infected. The closure of A is the set [ A ] = S ∞ t =0 A t of all vertices that are eventuallyinfected. When [ A ] = V we say that A percolates G , or simply that A percolates . We saythat A is closed under percolation if [ A ] = A .One would like to know under what conditions on G and p it is likely that A percolates G , so it is natural to define the critical probability p c ( G, r ) by p c ( G, r ) = inf { p : P p ([ A ] = V ( G )) > / } . (1)In the case G = Z d , by ergodicity (since the event that A percolates G is translationinvariant), the probability that A percolates G is either 0 or 1. Hence, on G = Z d , inequation (1) it is more natural to consider P p ([ A ] = Z d ) = 1 instead of P p ([ A ] = Z d ) > / d = r = 2 that for every positive initial density p there is percola-tion almost surely, and hence that p c ( Z ,
2) = 0. This was later greatly generalized bySchonmann [25], who showed that p c ( Z d , r ) = ( r d, d + 1 r d. (The cases r = 1 and d + 1 r d are trivial; the content of the theorem is the assertionwhen 2 r d .)The results of van Enter and Schonmann to a large extent ended the study of bootstrappercolation on infinite lattices. However, Aizenman and Lebowitz [1] recognized thatbootstrap percolation exhibited interesting finite-size effects: on finite grids [ n ] d , there is acertain metastability threshold for the initial density p , below which with high probabilitythere is no percolation, and above which with high probability there is percolation. Moreprecisely, Aizenman and Lebowitz showed that p c ([ n ] d ,
2) = Θ (cid:0) (log n ) − ( d − (cid:1) . Holroyd[18] later proved that p c ([ n ] ,
2) = (1 + o (1)) π /
18 log n , and Gravner, Holroyd and Morris[16] and Morris [22] obtained bounds on the second order term. Cerf and Cirillo [10]( d = r = 3) and Cerf and Manzo [11] ( d > r >
3) determined p c ([ n ] d , r ) up to a constantfor all r >
3, and Balogh, Bollob´as and Morris [4] ( d = r = 3) and Balogh, Bollob´as,Duminil-Copin and Morris [3] ( d > r >
3) determined the constant for all r > Z d and other lat-tices that have so far been studied have been shown to have critical probabilities on theappropriate infinite lattice equal to either 0 or 1. These include the r -neighbour modelon Z d , the r -neighbour model on general lattices embedded in Z d studied by Gravnerand Griffeath [15], the Duarte model studied by Schonmann [24] and Mountford [23], andnumerous other models (see, for example, [9, 29, 19]). In a recent paper, Bollob´as, Smithand Uzzell [8] introduced a new class of percolation models, called U -bootstrap percola-tion , which contains bootstrap percolation as a special case. They showed that many U -bootstrap percolation models on Z (those which they termed supercritical or critical )also have critical probabilities equal to zero. They also conjectured that the remainingmodels (those which they termed subcritical ) have strictly positive critical probabilities. Inthis paper we prove this conjecture. Together with the results in [8], this gives a completecharacterization of bootstrap-like models on Z that have non-trivial critical probabilities,under some natural assumptions listed in the next subsection. UBCRITICAL U -BOOTSTRAP PERCOLATION 3 U -bootstrap percolation. Under U -bootstrap percolation, new infections are madeaccording to any rule that is local (the rule depends on a bounded neighbourhood of thevertex), homogeneous (the same rule applies to every vertex) and monotone (the set ofneighbourhoods that infect a given site is an up-set). The formal definition is as follows.Let U = { X , . . . , X m } be a finite collection of finite, non-empty subsets of Z d \ { } andlet A = A ⊂ Z d . Then for each t >
0, let A t +1 = A t ∪ (cid:8) x ∈ Z d : there exists i ∈ [ m ] such that X i + x ⊂ A t (cid:9) . The set U is called an update family and the sets X i update rules . The r -neighbour modelon Z d is clearly an example of a U -bootstrap percolation model: it consists of (cid:0) dr (cid:1) updaterules, one for each r -subset of the neighbours of the origin. We again write [ A ] for theset of all vertices that eventually become infected, and say that A is closed under U if wehave [ A ] = A .For the rest of the paper we shall restrict our attention to the case d = 2. The roughbehaviour of two-dimensional U -bootstrap percolation is determined by the action of thedynamics on discrete half planes. We use the notation S for the unit circle in R and foreach u ∈ S we let H u denote the discrete half plane { x ∈ Z : h x, u i < } . An element u ∈ S is said to be a stable direction for the update family U if [ H u ] = H u ; that is, if nonew sites become infected when the initial set is equal to the half plane H u . Otherwise u is said to be an unstable direction for U . For every update family U and every u ∈ S , theclosure of H u is either H u or the whole plane Z . The stable set S for U is the set S = S ( U ) = { u ∈ S : u is stable for U } . We say that an update rule X destabilizes a direction u ∈ S if for U = { X } we have u / ∈ S ( U ). One can easily show (see Theorem 1.10 of [8]) that a subset S of the circle S is the stable set of some update family U if and only if S can be expressed as a finiteunion of closed intervals in S whose end-points have rational or infinite slope relative tothe standard basis vectors.Let T = R / π Z . We shall frequently need to change between elements of S andelements of T ; in order to do this we define the natural bijection u : T → S by u ( θ ) =(cos θ, sin θ ), and we set θ = u − to be its inverse function.We define the strongly stable set Int S ( U ) for U to be the interior of S , i.e.,Int S = Int S ( U ) = { u ∈ S : ∃ ε > | θ ( u ) − θ ( v ) | < ε then v ∈ S} . If u ∈ Int S then we say that u is a strongly stable direction . Clearly, any strongly stabledirection is also a stable direction.Bollob´as, Smith and Uzzell divided U -bootstrap percolation models into three classesaccording to the structure of the stable set. They defined the update family U to be:(i) supercritical if there exists an open semicircle in S that is disjoint from S ; thatis, if there do not exist three stable directions u , u and u such that the originbelongs to the interior of the triangle with vertices at u , u and u ;(ii) critical if every open semicircle in S has non-empty intersection with S , butthere exists a semicircle in S that is disjoint from Int S ; that is, if there existthree stable directions u , u and u such that the origin belongs to the interior PAUL BALISTER, B´ELA BOLLOB ´AS, MICHA L PRZYKUCKI, AND PAUL SMITH of the triangle with vertices at u , u and u , but no such three strongly stabledirections exist;(iii) subcritical if every open semicircle in S has non-empty intersection with Int S ;that is, if there exist three strongly stable directions u , u and u such that theorigin belongs to the interior of the triangle with vertices at u , u and u .Analogously to r -neighbour bootstrap percolation, we define p c ( Z , U ) to be the infimumof those values of p for which percolation occurs almost surely under update family U . In[8] the authors show that if U is either supercritical or critical then p c ( Z , U ) = 0. In fact,they show considerably more: letting p c ( Z , U , t ) = inf (cid:8) p : P p (0 ∈ A t ) > / (cid:9) , they show that p c ( Z , U , t ) = t − Θ(1) when U is supercritical and p c ( Z , U , t ) = (log t ) − Θ(1) when U is critical. (Considerably stronger results for critical models have since beenproved by Bollob´as, Duminil-Copin, Morris and Smith [6].) They also conjecture that p c ( Z , U ) > U is subcritical. Here we prove that conjecture. The following is themain theorem of this paper. Theorem 1.
Let U be a subcritical update family and let A ∼ Bin( Z , p ) . Then P p (cid:0) ∈ [ A ] (cid:1) → as p → . In particular, p c ( Z , U ) > . Furthermore, p c ( Z , U ) = 1 if and only if S = S . The strength of Theorem 1 lies in its generality: we prove that the critical probabilityis strictly positive for every two-dimensional bootstrap-like model for which the criticalprobability has not already been shown to be equal to zero.As previously remarked, Theorem 1 was previously only known in a small numberof exceptional cases, all of which we consider to be degenerate because they exhibit acertain symmetry property which trivializes the proof. We discuss these models further inSection 1.4.Combined with the results of [8], Theorem 1 has the following corollary.
Corollary 2.
Let U be an update family. Then p c ( Z , U ) > if and only if U is subcritical. Thus, our main theorem allows us to characterize all update families with non-trivialcritical probabilities.1.3.
The archetypal example: bootstrap percolation on the directed triangularlattice.
Let ~ T denote the triangular lattice embedded in C , oriented and scaled so that0 and 1 are neighbouring vertices. Let the edges of the lattice be directed, for k = 0 , , e (2 k +1) πi/ . In the resulting directed graph ~ T = ( V, E ), edges around anygiven vertex alternate in-out. (See Figure 1.)Let A = A ∼ Bin (cid:0) V ( ~ T ) , p (cid:1) , and for each integer t >
0, define the set of infected sitesat time t + 1 to be A t +1 = A t ∪ (cid:8) v ∈ V : | N − ( v ) ∩ A t | > (cid:9) , where N − ( v ) is the set of in-neighbours of v (that is, the set of vertices u neighbouring v such that −→ uv is an edge). Note that r = 2 is the only interesting value of the infectionthreshold for this model. We shall refer to this model as Directed Triangular BootstrapPercolation (DTBP). It is easy to see by coupling that p c ( ~ T ,
2) is at most the critical
UBCRITICAL U -BOOTSTRAP PERCOLATION 5 Figure 1.
Directed triangular lattice ~ T .probability for site percolation on T , the undirected triangular lattice, which is p sc ( T ) =1 /
2. (See Theorem 17 in [7].) Indeed, by the uniqueness of the infinite cluster in percolationon T , if we initially infect the vertices of ~ T with probability p > p sc ( T ) then almost surely allinitially healthy clusters of sites will be finite, and any such region is eventually infectedby the dynamics. However, it is not obvious whether p c ( ~ T ,
2) is strictly positive. It isknown that p c ( T ,
3) = 0 (see, e.g., [15]) but there is no apparent coupling between thetwo models that we could use to deduce anything about the critical probability in the2-neighbour bootstrap process on ~ T .However, by skewing the lattice ~ T , one can see that DTBP is equivalent to U -bootstrappercolation with update family U = { X , X , X } , where X = { (1 , , (0 , } , X = { ( − , − , (0 , } and X = { ( − , − , (1 , } . (See Figure 2.) Since U is subcritical,Theorem 1 implies that 0 < p c ( ~ T , < . By analysing carefully the proof of Theorem 1, one can in fact prove the following boundsfor p c ( ~ T , Corollary 3.
Under the DTBP subcritical U -bootstrap percolation model we have − < p c ( ~ T ,
2) = p c ( Z , U ) . . The upper bound in Corollary 3 is obtained by noting that DTBP can be coupled withoriented site percolation (see, for example, [17, 2]). Indeed, U -bootstrap percolation withupdate family U = { X } is precisely oriented site percolation: a site v remains healthyforever if and only if there exists an infinite up/right path starting at v of initially healthysites. The coupling with U gives p c ( ~ T , − p sc ( ~ Z ) . p sc ( ~ Z ) is thecritical probability for oriented site percolation on Z , and the final inequality is due toGray, Wierman and Smythe. For more information about percolation, see the book byBollob´as and Riordan, [7].Computer experiments suggest that the true value of p c ( ~ T ,
2) is far from both theupper and lower bound in Corollary 3, indicating that in fact p c ( ~ T , ∼ . PAUL BALISTER, B´ELA BOLLOB ´AS, MICHA L PRZYKUCKI, AND PAUL SMITH
Figure 2.
The equivalence of the update family U and the DTBP model;the dark grey site becomes infected when at least two of the light grey onesare. Figure 3.
The stable set S for the update family U (thick line); notethat indeed every semicircle in S intersects Int S .1.4. Symmetric models.
Apart from oriented site percolation, other previously studiedsubcritical U -bootstrap percolation models include the model U = (cid:8) { (1 , , (0 , } , { ( − , , (0 , − } (cid:9) , studied by Schonmann [24]; the knights, spiral and sandwich models, studied by Biroliand Toninelli [27] and by Jeng and Schwarz [20]; and the force-balance models, studied byJeng and Schwarz [21]. We would like to emphasize that none of these models is ‘typical’of the general model we study in this paper, in the following specific sense.Let us say that a (necessarily subcritical) model U is symmetric if the following propertyholds: there exists u ∈ S such that { u, − u } ⊂ Int S ( U ). It is easy to verify that all of theexamples in the previous paragraph are symmetric. Now if U is symmetric, then one cancouple U -bootstrap percolation from below with oriented site percolation, which gives anessentially trivial proof of Theorem 1 in the case of such models. We present this shortand elementary proof in Section 6.In general, however, subcritical models need not be symmetric (DTBP is not symmet-ric, for example), and in these cases there does not seem to be a useful coupling withoriented site percolation. For such models, the lack of symmetry makes it considerablyharder to control the growth of infected regions of sites, and the proof of Theorem 1 iscorrespondingly more complex. Thus, the non-symmetric models are the ones that weconsider to be ‘typical’.1.5. Organization of the paper.
The rest of this paper is organized as follows. In thenext section we give an outline of the proof of Theorem 1, and we explain heuristically whyone might expect the definition of a subcritical family to be the correct one. Followingthat, in Section 3, we set out the standard notation we shall use, and we formalize some
UBCRITICAL U -BOOTSTRAP PERCOLATION 7 of the definitions relating to our construction. In Section 4 we define and establish certainproperties of “barriers” and “triangular covers”, which will form the backbone of thecoupled process we use in the proof of Theorem 1. In Section 5 we assemble the varioustools from the previous sections in order to prove Theorem 6, which is a certain statementabout the existence of the “triangular covers”, and which should be thought of as theheart of Theorem 1. We then deduce Theorem 1 from Theorem 6. We end the paper firstwith Section 6, in which we point out that Theorem 1 is trivial if the update family U isassumed to be symmetric, and second with Section 7, in which we discuss a range of openproblems and conjectures. 2. Outline of the proof
We know that supercritical and critical families have critical probability in Z equal to0, so what is special about subcritical families that makes them behave differently? Let A be an initial set consisting of a rectangle of width m and arbitrary height, and a density p of sites above the rectangle. Under the classical two-neighbour bootstrap process on Z (which in a certain sense is representative of the behaviour of all critical processes), theinfection spreads upwards from the rectangle, filling every line completely until it meetsa fully healthy double line. The expected number of full new rows infected in the processis about (1 − p ) − m . The key property here is that a single site just above a full row willinfect all other sites on the same row. In other words, if R is the rectangle and x a sitenext to its upper edge then under the two-neighbour process there is no upper bound on | [ R ∪ { x } ] | − | R | that is uniform in m .Now consider the behaviour of the bootstrap process under an update family U forwhich u ( π/
2) is a strongly stable direction, that is, there is an interval of stable directionsaround u ( π/ A as in the previous paragraph, how many new sites do weexpect the process to infect? The key is that new sites create only localized infection: theset of additionally infected sites in the closure of the union of the rectangle and a smallset B of infected sites just above the top edge necessarily has “small” size, which dependson the size of B , on the stable set and on some additional characteristics of U , but not onthe size of the rectangle. Given B we can find a small circumscribed triangle T of B , withsides of T perpendicular to some stable directions within the interval of stable directionsaround u ( π/ u (0), u ( π ) and u (3 π/
2) are also stable directions, if theslopes of T are chosen appropriately to avoid the complications arising from the forbiddendirections which we define in Section 3.2, we have [ R ∪ B ] ⊂ [ R ∪ T ] = R ∪ T .The definition of a subcritical family is as follows: there exist three strongly stabledirections u , u and u such that the origin belongs to the interior of the triangle withvertices at u , u and u . Let H u,a denote the shifted half-plane { x ∈ Z : h x − a, u i < } .Then the condition that the origin lies inside the triangle with vertices at u , u and u implies that the triangular sets of the form T i =1 H u i ,a i , where the a i are arbitrary pointsin R , are necessarily finite. Also, we have [ T i =1 H u i ,a i ] = T i =1 H u i ,a i . In Section 3.2we show how to choose u , u and u so that these triangular sets are “robust” in thesense that they are still closed under U if we slightly perturb their edges, making them alittle bit “wiggly”. This is quite unlike the two-neighbour process, where the only finite PAUL BALISTER, B´ELA BOLLOB ´AS, MICHA L PRZYKUCKI, AND PAUL SMITH connected stable sets are rectangles, and new sites on their edges cause entire new rowsor columns of infection.In our proof of Theorem 1 we exploit the above property of subcritical update families.We show that if every site in Z is initially infected independently with some probability p > p is small enough, almost surely one can find a collection of slightly perturbedtriangles (as above) with the following properties: • every eventually infected site is contained in at least one triangle, • if two triangles have a nonempty intersection or, in fact, if they are not wellseparated, then one of them is contained in the other, • any site in Z belongs to at least one triangle with probability tending to 0 as p → p , the existence of a collection of triangular sets with these propertiesproves that the initial set does not percolate the plane, and this implies the lower boundon p c ( Z , U ) in Theorem 1.We find our collection of perturbed triangles using a renormalization argument. Ourmethod is motivated by the techniques introduced by G´acs [13] in the context of clairvoyantscheduling and a certain equivalent dependent oriented percolation model. We partitionthe plane using successively coarser tilings into squares of side lengths ∆ ≪ ∆ ≪ . . . .At each scale ∆ i we will have a notion of an ( i ) -good ∆ i -square, where “good” will roughlycorrelate with “being sparsely infected”, and there will be a corresponding notion of an( i ) -bad ∆ i -square. A little more precisely, a ∆ i -square will be ( i )-good if all ( i − i − -squares contained in it and in its close neighbourhood are quite strongly isolated.Inductively we show that an ( i )-bad ∆ i -square contained in a ( i + 1)-good ∆ i +1 -squarecan be enclosed in a perturbed triangle which is not too large and is well separated, forall j i , from all ( j )-bad ∆ j -squares which are not fully contained in it. Additionally,this perturbed triangle has sides essentially perpendicular to stable directions u , u and u , i.e., is on its own closed under U . We do this by showing simultaneously by inductionthat, for any i , one can always find a “thick” healthy barrier through ( i )-good ∆ i -squares,disjoint from ( j )-bad squares for all j < i . Since an ( i )-bad ∆ i -square contained in an( i + 1)-good ∆ i +1 -square is necessarily surrounded by ( i )-good ∆ i -squares, this allows usto construct the triangular sets which enclose our eventually infected area.The main task is the second part of the induction: to show that one can constructbarriers through ( i )-good ∆ i -squares. The idea is that, since all ( i − i − -squarescontained in an ( i )-good ∆ i -squares are quite strongly isolated, it is possible to “navigatearound” these ( i − i − -squares without straying too far from a straight line, andto use the induction hypothesis to construct the barrier through the ( i )-good ∆ i -squaresout of consecutive sub-barriers through ( i − i − -squares.In order to be a little more precise, suppose we are trying to construct a healthy barrierbetween sites x and y , where these are such that the line ℓ joining them is roughly perpen-dicular to u and only passes through ( i )-good ∆ i squares. We shall show that there existcertain “(i)-clean sites” c , . . . , c k , all of which lie close to ℓ , such that the union of thelines joining x to c , c to c , and so forth, up to c k to y , only passes through ( i − i − -squares. By induction, it follows that there exists a healthy barrier joining x to c ,etc., and one can show that it is possible to control these sufficiently such that their union UBCRITICAL U -BOOTSTRAP PERCOLATION 9 is again a healthy barrier, but at the next scale. Thus, the edges of the perturbed trianglesthat we construct are in fact perturbed at all scales .This is the only part of the proof where we use the subcriticality of the update familyand for that reason it is the most important part of our argument. The assertion thatone can always find these perturbed triangles is Theorem 6, and the (key) sub-assertionthat one can always find these healthy barriers is Lemma 7: these two results should beregarded as the heart of Theorem 1.3. Additional notation and definitions
Notation.
Given two sites a, b ∈ Z we define dist( a, b ) = k a − b k . For any two sets A, B ⊂ Z we then take dist( A, B ) = min a ∈ A, b ∈ B dist( a, b ) . For an update family U we define ∇ ( U ) = max i ∈ [ m ] max a,b ∈ X i dist( a, b ) . Hence, in particular, if A is a set of initially infected sites such that any two distinct sitesin A are at distance larger than ∇ ( U ) then under update family U we have [ A ] = A .Given two sites a, b ∈ Z , a = b , let u a,b = b − a dist( a, b ) ∈ S . Subcritical update families are those for which there exist three strongly stable directions u , u and u such that the origin belongs to the interior of the triangle with vertices at u , u and u . This can be rephrased as: there exist three distinct stable directions u , u , u and positive numbers λ , λ , λ , ε > λ u + λ u + λ u = 0 , (2)(ii) for t = 1 , , { u : | θ ( u t ) − θ ( u ) | < ε } ⊂ S . (3)To simplify our proof we will, somewhat counterintuitively, take the ε in (3) to be verysmall (which we are of course free to do).3.2. Choice of strongly stable directions and the first bound on ε . In this sectionwe choose our strongly stable directions u , u and u , and we give a first upper boundon ε in (3). The reason why we impose these particular conditions on our parameters willbecome clear in the proof of Lemma 4 in Section 4. Note that if u is a strongly stabledirection such that N ε ( u ) = { u : | θ ( u ) − θ ( u ) | < ε } ⊂ S then clearly also N ε ( u ) ⊂ Int S ,i.e., all directions in N ε ( u ) are strongly stable. This means that the existence of onetriple of strongly stable directions satisfying (2) implies the existence of infinitely manysuch triples.Given an update family U = { X , . . . , X m } , we say that a direction u ∈ S is forbiddenfor U if it is perpendicular to at least one side of the convex hull of at least one of theupdate rules X i (note that every side of any convex hull forbids 2 opposite directions). Let F ( U ) = { u : u is forbidden for U } be the set of directions forbidden for U . For example,for the update family U equivalent to DTBP introduced in Section 1.3 we have F ( U ) = ( √ , √ ! , − √ , − √ ! , − √ , √ ! , √ , − √ ! , − √ , √ ! , √ , − √ ! ) . Since F ( U ) is a finite set, we can choose our strongly stable directions u , u , u ∈ Int S ( U ) \ F ( U ) and let ε ( u , u , u ) be small enough so that for i = 1 , ,
3, we have N ε ( u ,u ,u ) ( u i ) ⊂ Int
S \ F ( U ) . (4)To simplify our proof, from now on we assume that in (3) we have ε ε ( u , u , u ).3.3. Good squares.
Let us now define more precisely the tilings of Z we will work within this paper, as well as the concepts of good and bad squares. The coarseness of our tilingsand the definitions of good and bad squares will depend on the following parameters. Let1 < γ < β < α < δ = (2 α + 2 β − / (2 − α ) . (5)Let { ∆ i } ∞ i =1 be an increasing sequence of natural numbers with∆ i +1 = min { n ∈ N : n > ∆ αi and n is a multiple of ∆ i } , with ∆ > ∇ ( U ) to be defined later. For i > q i = ∆ − δi , g i = ∆ βi , and σ i = ε/ ε ∆ − γi . Note that, since α > β >
1, we have ∆ i +1 > g i > ∆ i .For each i > i )-tiling of Z with ∆ i × ∆ i squares, i.e., a partitionof Z into sets of the form { a ∆ i + 1 , a ∆ i + 2 , . . . , ( a + 1)∆ i } × { b ∆ i + 1 , b ∆ i + 2 , . . . , ( b + 1)∆ i } for all a, b ∈ Z . Note that our ( i )-tilings are nested, i.e., that every ∆ i +1 × ∆ i +1 squareconsists of ⌈ ∆ α − i ⌉ squares of side length ∆ i .We shall define squares of side length ∆ i in our ( i )-tiling of Z to be either ( i ) -good or( i ) -bad . A ∆ -square is (1)-good if all its sites are initially healthy, otherwise it is (1)-bad.For i > S of side length ∆ i +1 to be ( i + 1)-bad if there exist twodistinct non-adjacent squares (we consider squares that only touch corners as adjacent) S ′ , S ′′ of side length ∆ i in our ( i )-tiling (where S ′ and S ′′ might be disjoint from S ) whichare ( i )-bad and such that max { dist( S, S ′ ) , dist( S, S ′′ ) , dist( S ′ , S ′′ ) } g i .For i > i )-good square S we say that a site v ∈ S is ( i ) -clean if, for all j < i , v is at distance at least g j / j )-bad square.4. Barriers and triangular covers
In this section we define barriers and triangular covers. We shall use these concepts inour proof to show that for p > Z , by showing that the closure of the initial infection can be enclosed in a collectionof separated, finite sets of a special triangular shape. UBCRITICAL U -BOOTSTRAP PERCOLATION 11 Recall that we assume that for our update family U we have [ t =1 { u : | θ ( u t ) − θ ( u ) | < ε } ⊂ S . (6)If for some t ∈ { , , } we have (cid:12)(cid:12)(cid:0) θ ( u x,y ) − θ ( u t ) (cid:1) (mod 2 π ) − π/ (cid:12)(cid:12) < σ , (roughly speaking, if u x,y is “nearly” perpendicular to the stable direction u t ), then a(1 , t ) -barrier joining x to y is the set of all sites v ∈ Z such that for some λ ∈ [0 ,
1] wehave dist( v, λx + (1 − λ ) y ) ∇ ( U ) . Let i > x, y ∈ Z be such that (cid:12)(cid:12)(cid:0) θ ( u x,y ) − θ ( u t ) (cid:1) (mod 2 π ) − π/ (cid:12)(cid:12) < σ i . Let, for some m >
1, the sequence ( z j ) mj =0 with z = x , z m = y and z j ∈ Z for all j = 1 , , . . . , m −
1, be such that for all j = 1 , , . . . , m we have (cid:12)(cid:12)(cid:0) θ ( u z j − ,z j ) − θ ( u t ) (cid:1) (mod 2 π ) − π/ (cid:12)(cid:12) < σ i − . Then the set of all sites v ∈ Z such that for some j ∈ { , , . . . , m } and some λ ∈ [0 , v, λz j − + (1 − λ ) z j ) ∇ ( U )is an ( i, t ) -barrier joining x to y (see Figure 4). The sequence ( z j ) mj =0 is called the anchor of the ( i, t )-barrier. Note that for i >
2, an ( i, t )-barrier consists of m > i − , t )-barrier. This compound structure will allow ( i, t )-barriersto avoid infected regions in Z . We shall later use such infection-avoiding barriers and,exploiting the fact that they are essentially perpendicular to stable directions, encloseinfected regions in hulls from which they cannot break out.We may assume that 0 θ ( u ) < θ ( u ) < θ ( u ) < π . Let K ⊂ Z be finite, let i > x, y, z ∈ Z are distinct points such that: • an ( i, x to y , an ( i, y to z and an ( i, z to x exist, and • K lies inside the area bounded by these barriers and is disjoint from them.Then we call the union B of the three barriers an ( i ) -barrier cover of K , and we call theunion T of B and the sites in the area bounded by B an ( i ) -triangular cover of K . Notethat there exist infinitely many ( i )-barrier covers and infinite many ( i )-triangular coversof any given finite set K for every i > i, t )-barriers are perpendicular to strongly stable directions. In the next lemmawe use this fact to show that for any finite set K , any i >
1, and any ( i )-barrier cover B and associated ( i )-triangular cover T of K , the closure [ K ] is a subset of T \ B and istherefore isolated from Z \ T by a barrier of thickness at least ∇ ( U ).Note that for any subcritical update family U = { X , . . . , X m } , for all 1 i m wehave | X i | >
2. Indeed if, without loss of generality, X = { ( x, y ) } , then every direction u ∈ S such that h ( x, y ) , u i < S and hence the family U is supercritical. Also, we then trivially u t z = x z z z z z = y Figure 4.
An example of an ( i, t )-barrier joining x to y with ∇ ( U ) = 2.have p c ( Z , U ) = 0: every site ( u, v ) ∈ Z will become infected if for some t > u, v ) + t · ( x, y ) is initially infected and this happens almost surely for any p > Lemma 4.
Let K ⊂ Z be finite, let i > , and let B be an ( i ) -barrier cover of K and T its associated ( i ) -triangular cover. Then [ K ] ⊂ [ T \ B ] = T \ B. Proof.
The first containment [ K ] ⊂ [ T \ B ] is obvious because K ⊂ T \ B . Therefore weonly need to prove that [ T \ B ] = T \ B , i.e., that T \ B is closed under U .Assume that the initial set of infected sites is T \ B . Recall that we have u , u , u and ε ε ( u , u , u ) in Section 3.2 such that for t ∈ { , , } we have N ε ( u ,u ,u ) ( u t ) ⊂ Int
S \ F ( U ).A site v ∈ Z \ ( T \ B ) can become infected for three, essentially different, reasons.These are schematically shown in Figure 5, where we assume that T \ B lies below thesolid curve. Cases (1) and (2) in Figure 5 correspond to v being infected using updaterules X ′ and X ′′ , which destabilize directions u ′ and u ′′ respectively. For simplicity weassume | X ′ | = | X ′′ | = 2. Case (3) corresponds to v being infected using an update rule X ′′′ that does not destabilize any directions. Rules of this type necessarily contain theorigin in their (closed) convex hull; in the figure, for simplicity, we assume | X ′′′ | = 3.The site v cannot be infected for the reason shown in case (1) of Figure 5, becausethe existence of such a rule X ′ ∈ U would contradict the fact that for t ∈ { , , } wehave N ε ( u ,u ,u ) ( u t ) ⊂ S . It cannot be infected for the reason shown in cases (2) or(3) of Figure 5, because now the existence of such a rule would contradict the fact that N ε ( u ,u ,u ) ( u t ) ∩ F ( U ) = ∅ . Hence T \ B is closed under U , which completes the proof. (cid:3) UBCRITICAL U -BOOTSTRAP PERCOLATION 13 ×× vu ′ (1) × × v u ′′ (2) × ×× v (3) Figure 5.
Three ways to infect a site v ∈ Z \ ( T \ B ). The sites in v + X ′ , v + X ′′ and v + X ′′′ are denoted by × .In the next lemma we show that there exists a constant c = c ( U ) such that for all i > i )-triangular cover of a square of side length ∆in a “small” neighbourhood of that square, i.e., in a larger square of side length at most c ∆. Lemma 5.
There exists ℓ ∈ N and ε > depending only on U such that the followinghold. Let ε ε , ℓ > ℓ , i > , and ∆ > ∇ ( U ) . Consider the tiling of [ c ∆] consistingof (2 ℓ + 1) squares of side length ∆ , where c = 2 ℓ + 1 . Then this tiling contains threedistinct ∆ × ∆ squares Y , Y and Y such that for all y ∈ Y , y ∈ Y and y ∈ Y , andfor each t = 1 , , , we have (cid:12)(cid:12)(cid:0) θ ( u y t ,y t +1 ) − θ ( u t ) (cid:1) (mod 2 π ) − π/ (cid:12)(cid:12) < ε/ , (7) where y = y .Additionally, every ( i, -barrier joining y to y , every ( i, -barrier joining y to y and every ( i, -barrier joining y to y , is contained within the tiling and is disjoint fromits middle square, i.e., from Y = [∆ ℓ + 1 , ∆( ℓ + 1)] × [∆ ℓ + 1 , ∆( ℓ + 1)] . Proof.
Let i > > ∇ ( U ), and let ℓ > v = (∆ ℓ + (∆ + 1) / , ∆ ℓ + (∆ + 1) / Y . The whole of Y is clearly contained in a circle of radius ∆ centeredat v . For r > S and S be the circles centered at v of radius r ∆and ( r + 3)∆ respectively. Also, let T and T be the triangles circumscribed on S and S respectively, tangent to these circles, for t = 1 , ,
3, at points v + r ∆ u t and v + ( r + 3)∆ u t respectively. (See Figure 6.) Independently of the values of u j and r , the three grey cornerregions in Figure 6 are each large enough to contain a disc of diameter 3∆, each of whichitself contains a ∆ × ∆ square of the tiling of [ c ∆] . Fix any such three squares Y , Y and Y . We claim that if r is large enough and ε > i ) then Y , Y and Y satisfy the conclusions of the lemma. u u u ( r + 3)∆ r ∆ v y Figure 6.
Finding ( i, t )-barriers in the neighbourhood of v .For t = 1 , ,
3, let θ t = θ ( u t ). Without loss of generality we may assume that, modulo2 π , we have θ − θ > θ − θ > θ − θ , and we may also assume that r >
3. The longestside of T has length a max = ( r + 3)∆ (cid:18) tan (cid:18) θ − θ (cid:19) + tan (cid:18) θ − θ (cid:19)(cid:19) r ∆ (cid:18) tan (cid:18) θ − θ (cid:19) + tan (cid:18) θ − θ (cid:19)(cid:19) , while the shortest side of T has length a min = r ∆ (cid:18) tan (cid:18) θ − θ (cid:19) + tan (cid:18) θ − θ (cid:19)(cid:19) . First we verify that (7) holds. Let y ∈ Y , y ∈ Y and y ∈ Y . For t = 1 , ,
3, wemust show that θ ( u y t ,y t +1 ) is at most ε/ u t (where again y = y ). This holds if a min tan (cid:16) ε (cid:17) > , (8)because this condition guarantees that the whole grey corner region containing Y t +1 iscontained inside the angle with its vertex at y t and of measure ε , lying symmetricallyaround the line perpendicular to u t which goes through y t . (See Figure 6 with t = 2.)Inequality (8) is satisfied whenever r > (cid:16) tan (cid:16) ε (cid:17)(cid:17) − tan (cid:18) θ − θ (cid:19) + tan (cid:18) θ − θ (cid:19) ! − = r ε . UBCRITICAL U -BOOTSTRAP PERCOLATION 15 Thus, (7) holds provided r > r ε .Finally we must show that the condition in the last paragraph of the lemma holds. Givenany two sites u and w in Z , and t ∈ { , , } , the sequence ( z j ) mj =0 of points forming theanchor of an ( i, t )-barrier joining u to w is, by the definition of an ( i, t )-barrier, containedin a rhombus with two of its vertices at u and w and the interior angles at these twovertices equal to 2 ε . Now, if u and w are contained in different grey corner regions inFigure 6, then one can easily verify that this rhombus is at distance at least r ∆ / v , provided a max tan ε r ∆ /
2, which holds if ε arctan tan (cid:18) θ − θ (cid:19) + tan (cid:18) θ − θ (cid:19) ! − = ε . Note that ε depends on the values of θ t only. Thus, to ensure that every ( i, t )-barrierjoining u and w is disjoint from the small circle centered at v , it is enough to have r ∆ / > ∇ ( U ), which is true whenever r > > ∇ ( U )).The assertion that the barriers are entirely contained within [ c ∆] if c is sufficiently largefollows immediately from the fact that by the choice of ε every point of every ( i, t )-barrieris at distance at most a max + r ∆ / ∇ ( U ) ∆ r tan (cid:18) θ − θ (cid:19) + tan (cid:18) θ − θ (cid:19) ! + r ! from v . Therefore, for ε ε and r = max { , r ε } the lemma holds with ℓ = & r tan (cid:18) θ − θ (cid:19) + tan (cid:18) θ − θ (cid:19) ! + r ' . (cid:3) Given a set of stable directions S and our choice of strongly stable and not forbiddendirections u , u and u in Section 3.2, let c ( S ) be the smallest c = 2 ℓ + 1 for which Lemma5 holds for ε = min { ε , ε ( u , u , u ) } .Now let K ⊂ Z be finite and let ∆ > K is contained in asquare of side length ∆. We say that an ( i )-triangular cover for a finite set K is tight if itis completely contained in the (cid:0) c ( S )∆ (cid:1) × (cid:0) c ( S )∆ (cid:1) square centered at any minimal square(necessarily of side length ∆) containing K .5. Positive critical probability: The proof of Theorem 1
The aim of the first part of this section is to state and prove the theorem that willbe our main tool in proving Theorem 1. We described the outline of the proof of thistheorem in Section 2. Before stating the theorem, we need a few preliminary definitionsand remarks.For each k >
1, we say that the measure P p is ( k, -independent if, for every pair ofnon-adjacent squares S and T of side length ∆ k in the ( k )-tiling, the events { S is ( k )-good } and { T is ( k )-good } are independent.Recall that, for i >
1, a site v in an ( i )-good square is said to be clean if, for all j < i , v is at distance at least g j / j )-bad square. In the statement of the theorem we refer to unions of pairwise adjacent ( i )-bad ∆ i -squares. Note that at most four such squares can be all pairwise adjacent and that aunion of such squares is always contained in a 2∆ i × i square.After the initial infection is seeded, bootstrap percolation is a fully deterministic process.Hence, given a set of initially infected sites A ⊂ Z , for each k > X k be the collectionof all sets X ⊂ Z such that X is a union of pairwise adjacent ( k )-bad squares and X intersects a ( k + 1)-good square. Theorem 6.
Let U be a subcritical update family with three strongly stable directions u , u , u ∈ Int
S \ F ( U ) such that for some positive numbers λ , λ , λ we have λ u + λ u + λ u = 0 . Then, if p > is small enough, for each k > the following threeconditions hold:(i) The measure P p is ( k, -independent, and for any ∆ k × ∆ k square S in the ( k ) -tilingwe have P p ( S is ( k ) -bad ) q k . (ii) Every ( k ) -good square S contains a ( k ) -clean site.(iii) For every X ∈ X k there exists a tight ( k ) -triangular cover T k ( X ) such that, for distinct Y, Z ∈ X k , the sets T k ( Y ) and T k ( Z ) are disjoint, and for each i < k , if Y ∈ X k and Z ∈ X i , then either T k ( Y ) and T i ( Z ) are disjoint or T i ( Z ) ⊂ T k ( Y ) . Since all ( k )-triangular covers we consider henceforth will be tight, we shall alwaysassume that this extra condition is understood, and make no further mention of it. Proof.
Given a choice of α, β, γ and δ satisfying (5), and ε = min { ε , ε ( u , u , u ) } where ε is taken as in the proof of Lemma 5 and ε ( u , u , u ) as in Section 3.2, let ∆ be largeenough to satisfy the following five conditions: • ∆ > max { δ +5 , ∇ ( U ) } , • ∆ α − > c ( S ) , • ∆ β − > max { , c ( S ) } , • ∆ α − β > , • ∆ β − − γ > c ( S ) /ε. Let sites in Z be initially infected independently with probability p = (∆ ) − δ − . We shallprove Theorem 6 by induction on k >
1. First we check the case k = 1.(i) Any ∆ -square A is (1)-good if it is initially fully healthy. Thus we immediately seethat states of all ∆ -squares are mutually independent. We also have P p ( A is (1)-bad) < (∆ ) p = (∆ ) (∆ ) − δ − = (∆ ) − δ = q . (ii) Every site in a (1)-good ∆ -square is (1)-clean (the condition of a (1)-clean site isempty) and therefore Condition (ii) is trivially satisfied by any (1)-good ∆ -square.(iii) For k = 1 Condition (iii) is empty and is therefore trivially satisfied by any (1)-good∆ -square.Assume now that the three conditions of Theorem 6 are satisfied by our ( i )-tilings forall 1 i k . Let us consider the ( k + 1)-tiling of Z .(i) The state of any square X in our ( k + 1)-tiling (either “( k + 1)-good” or “( k + 1)-bad”)depends only on the states of squares in the ( k )-tiling within distance g k = ∆ βk of X . UBCRITICAL U -BOOTSTRAP PERCOLATION 17 If ∆ α − β > k we have g k ∆ k +1 / k +1 -squares Y and Z depend on states of non-adjacent sets of ∆ k -squares. Byinduction, the states of squares in these non-adjacent sets are independent. Thereforethe states of Y and Z are independent. Hence the states of all non-adjacent ∆ k +1 -squares are independent.If a ∆ k +1 × ∆ k +1 square S is ( k +1)-bad then it contains or is at distance at most g k from two non-adjacent ( k )-bad squares X, Y in our ( k )-tiling such that dist( X, Y ) g k . Hence, given S , there are at most (cid:18) ∆ k +1 + ∆ k + 2 g k ∆ k (cid:19) ways of choosing X and then, assuming that Y is contained in the semicircle of radius g k below X , we have 2( g k / ∆ k ) ways of choosing Y . Recall that ∆ k +1 < ∆ αk + ∆ k and that for all k > k > ∆ > δ +5 . Since q k = ∆ − δk , where δ =(2 α + 2 β − / (2 − α ), and the states of non-adjacent squares are independent, wehave P p ( A is ( k + 1)-bad) < (cid:18) ∆ k +1 + ∆ k + 2 g k ∆ k (cid:19) (cid:18) g k ∆ k (cid:19) q k < (cid:18) k +1 ∆ k (cid:19) (cid:16) ∆ β − k (cid:17) ∆ − δk < (cid:0) α − k (cid:1) ∆ β − − δk = 2 δ ∆ − k − δ ∆ α +2 β − − δk − δ ∆ α +2 β − − δk = (2∆ αk ) − δ q k +1 . (ii) If a ( k + 1)-good square S does not contain any ( k )-bad subsquare then, in particular,any square Y in our ( k )-tiling contained in the middle ∆ k +1 / × ∆ k +1 / S is ( k )-good and lies at distance at least ∆ k +1 / > g k / k )-bad square.Since Y is ( k )-good it contains a ( k )-clean site v . Since v is at distance at least g k / k )-bad square, v is also ( k + 1)-clean.Hence assume that S contains a ( k )-bad square X . Since S is ( k + 1)-good, anyother ( k )-bad square within distance g k of X (not necessarily contained in S ) must beadjacent to X . It follows that, since ∆ β − >
30, every site at distance between 2 g k / g k / X is at distance at least g k / k )-bad square. At least aquarter of the ring of sites at distance between 2 g k / g k / X lies inside S . Additionally, this ring is thick enough to contain a 3∆ k × k square, which itselfcontains a ( k )-good square with a ( k )-clean site v . By the same argument as in theprevious paragraph, v is also ( k + 1)-clean.(iii) Consider a ( k + 1)-good square S and a union X of pairwise adjacent ( k )-bad squaresintersecting S (as usual, X is contained within a 2∆ k × k square). By Lemma 5, thedefinition of a ( k + 1)-good square, and since ∆ β − > c ( S ), the 2 c ( S )∆ k × c ( S )∆ k square C centered at X does not intersect with the 2 c ( S )∆ k × c ( S )∆ k square centered at any other union of adjacent ( k )-bad squares. Additionally, C contains three ( k )-good squares C , C and C with ( k )-clean sites c ∈ C , c ∈ C and c ∈ C suchthat all ( k, c to c , all ( k, c to c and all ( k, c to c are contained within C . Also, these barriers are disjoint from X , which lies inside the area bounded by them.Therefore we need to prove that between any two of c , c and c we can findappropriate barriers avoiding T i ( Y ) for any union Y of adjacent ( i )-bad squares forall i < k . Then the union of these three barriers and the area inside them will be ourdesired T k ( X ), the ( k )-triangular cover of X . To do this we shall prove the followingcrucial lemma. We would like to emphasize that this lemma is the key to the thirdand most important part of Theorem 6. The theorem follows from the lemma in anessentially straightforward way. Lemma 7.
Let j > . Let x and y be two ( j ) -clean sites in different ( j ) -goodsquares such that for some t ∈ { , , } we have (cid:12)(cid:12)(cid:0) θ ( u x ,y ) − θ ( u t ) (cid:1) ( mod π ) − π/ (cid:12)(cid:12) < σ j = ε/ ε/ ∆ γj . Suppose also that all ∆ j × ∆ j squares in our ( j ) -tiling within distance ∆ j of thesegment with x and y as endpoints are ( j ) -good. Then there exists a ( j, t ) -barrierjoining x to y that does not intersect the ( i ) -triangular cover T i ( X ) of any union X of neighbouring ( i ) -bad squares for any i < j .Proof. For j = 1 the assertion is empty and so the lemma is trivial. Thus assumethat the lemma holds for j m . Let x and y be two ( m + 1)-clean sites in different( m + 1)-good squares such that, for some t ∈ { , , } , (cid:12)(cid:12)(cid:0) θ ( u x ,y ) − θ ( u t ) (cid:1) (mod 2 π ) − π/ (cid:12)(cid:12) < σ m +1 holds. Recall that every ( m + 1)-clean site is also ( m )-clean.Let x = x + 8 c ( S )∆ m u (cid:0) θ ( u x,y ) + π/ (cid:1) ,y = y + 8 c ( S )∆ m u (cid:0) θ ( u x,y ) + π/ (cid:1) ,x = x + 8 c ( S )∆ m u (cid:0) θ ( u x,y ) − π/ (cid:1) ,y = y + 8 c ( S )∆ m u (cid:0) θ ( u x,y ) − π/ (cid:1) , and for ℓ = 0 , , Z ℓ = { v ∈ Z : dist( v, λx ℓ + (1 − λ ) y ℓ ) c ( S )∆ m for some λ ∈ [0 , } (see Figure 7).If ∆ m +1 > c ( S )∆ m , which is true since ∆ α − > c ( S ), then S ℓ =1 Z ℓ is containedin a union of ( m + 1)-good squares. This implies that every union of pairwise adjacent( m )-bad squares intersecting S ℓ =1 Z ℓ is at distance at least g m from any other ( m )-badsquare. Additionally, the ( m )-triangular cover of any union X of pairwise adjacent( m )-bad squares, being contained in the 2 c ( S )∆ m × c ( S )∆ m square centered at X ,intersects at most two of the sets Z ℓ .Assume that S ℓ =1 Z ℓ intersects d such 2 c ( S )∆ m × c ( S )∆ m squares containingunions of adjacent ( m )-bad squares: Y , Y , . . . , Y d , ordered according to their distance UBCRITICAL U -BOOTSTRAP PERCOLATION 19 from x . For every s ∈ [ d ], let y s ∈ R be the centre of Y s and let ℓ s ∈ { , } be anindex of a set Z ℓ that is avoided by Y s . Then in Z ℓ s we can find an ( m )-good square C s at distance at least 4 c ( S )∆ m and at most 6 c ( S )∆ m from Z , with an ( m )-cleansite z s ∈ C s , such that the distance between z s and the line going through x , x and x differs from the distance between y s and that line by at most ∆ m . Note that theconditions on the location of C s imply that C s is at distance at least 3 c ( S )∆ m / Z \ Z ℓ s . See Figure 7 for a graphical interpretation of this description. x y Z x y Z x y Z Y Y z z Figure 7.
The ( m )-clean sites z and z used to bypass unions of adjacent( m )-bad squares Y and Y and to inductively construct an ( m +1 , t )-barrierjoining x to y .Set z = x and z d +1 = y . Note that if the segment joining z s to z s +1 is at distanceat least ∆ m from any ( m )-triangular cover of any union of adjacent ( m )-bad squares(this clearly implies that the segment is at distance at least ∆ m from any ( m )-badsquare) and if (cid:12)(cid:12)(cid:0) θ ( u z s ,z s +1 ) − θ ( u t ) (cid:1) (mod 2 π ) − π/ (cid:12)(cid:12) < σ m , then by the induction hypothesis there exists an ( m, t )-barrier joining z s to z s +1 satisfying the lemma. If this holds for all pairs of consecutive z s s then these ( m, t )-barriers together constitute an ( m + 1 , t )-barrier joining x to y which avoids, for all i m , ( i )-triangular covers of all unions of neighbouring ( i )-bad squares.Since ∆ β − >
30, using the bound arcsin φ πφ/ φ ∈ [0 , θ ( u z s ,z s +1 ) and θ ( u x ,y ) modulo 2 π is bounded from above byarcsin (cid:18) c ( S )∆ m g m − m (cid:19) < πc ( S )∆ m g m < ε γm for all m > β − − γ > c ( S ) /ε . Since σ m − σ m +1 = ε ∆ γm − ε ∆ γm +1 > ε γm , we see that for ∆ > (68 c ( S ) /ε ) / ( β − − γ ) the angles between consecutive z s s allow usto find ( m, t )-barriers between these sites.Let us then show that the segment joining z s to z s +1 is at distance at least ∆ m from any ( m )-triangular cover of any union of adjacent ( m )-bad squares. First, weobserve that z s and z s +1 are at distance at least 3 c ( S )∆ m / Z \ S ℓ =1 Z ℓ , so wedo not need to consider ( m )-bad squares lying outside S ℓ =1 Z ℓ .We chose z s to be at distance at least 4 c ( S )∆ m from Z , and consequently alsofrom Y s . Let w ′ be a site in a ( m )-triangular cover of Y s . Then, by Lemma 5, thedistance between w and the line going through x , x and x is not larger than thedistance between z s and this line by more than 2 c ( S )∆ m . Let w ′′ be a point in thesegment joining z s to z s +1 at distance at most 2 c ( S )∆ m from Z . If ε/ (2∆ γm ) π/ ε π/
4, then w ′′ is at distance from the line going through x , x and x larger by at least 4 c ( S )∆ m than z s is. Therefore, the segment joining z s to z s +1 is at distance at least 2 c ( S )∆ m from Y s and so at distance at least ∆ m from T m ( Y s ). In a similar way we show that it is at distance at least ∆ m from T m ( Y s +1 ).By the choice of the ordering of the squares Y s we know that no other ( m )-triangularcover of any union of adjacent ( m )-bad squares is near the segment joining z s to z s +1 and the lemma is proved. (cid:3) From Lemma 5 and Lemma 7 it follows immediately that for any union X ofadjacent ( k )-bad squares inside a ( k + 1)-good square we can find a ( k )-triangularcover T k ( X ) of X inside the 2 c ( S )∆ k × c ( S )∆ k square centered at X , satisfying therequirements of Theorem 6.This completes the proof of the theorem. (cid:3) In the next lemma we show that the collection of triangular covers, which by Theorem6 almost surely exists if p > Z that everbecomes infected.Recall that X k is the collection of all sets X ⊂ Z such that X is a union of pairwiseadjacent ( k )-bad squares and X intersects a ( k + 1)-good square. Lemma 8.
Given a subcritical family U , let p = (∆ ) − δ − > be small enough so thatTheorem 6 holds. Let A ∼ Bin( Z , p ) . Then, almost surely, [ A ] ⊂ Z = [ i > [ X ∈X i T i ( X ) . Proof.
By the definition of the closure, the set [ A ] is the smallest set that contains A andis closed under U .We show first that A ⊂ Z . Note that since we define q i = ∆ − δi ∆ − α i − δ , we have P i > q i < ∞ . Every ∆ -square that contains at least one initially infected site is (1)-badand, by the Borel-Cantelli lemma, P i > q i < ∞ implies that every site in Z is containedin infinitely many good squares almost surely. In particular, every initially infected sitewill be contained in the triangular cover of a union of adjacent ( i )-bad squares intersectingan ( i + 1)-good square, for some i >
1. Thus to prove the lemma we just need to showthat Z is closed under U .As shown in Lemma 4, for any i > i )-triangular cover of any union X of adjacent( i )-bad squares is closed under U . Moreover, the infected interior of the cover is separated UBCRITICAL U -BOOTSTRAP PERCOLATION 21 from Z \ T i ( X ) by a healthy barrier of thickness at least ∇ ( U ). By condition (iii) inTheorem 6, for all i > j >
1, any union X of adjacent ( i )-bad squares and any union Y of adjacent ( j )-bad squares satisfy either T j ( Y ) ⊂ T i ( X ) or T j ( Y ) ∩ T i ( X ) = ∅ . Hence,by the definition of ∇ ( U ), any collection of triangular covers is closed under U and, inparticular, so is Z . This means that [ A ] ⊂ Z and the proof of the lemma is complete. (cid:3) Equipped with Theorem 6 and Lemma 8, we are now in a position to prove Theorem 1.
Proof of Theorem 1.
Having proved Theorem 6 and Lemma 8, to prove the inequality p c ( Z , U ) > p > i > X of adjacent ( i )-bad squares such that the site (0 , c ( S )∆ i × c ( S )∆ i square centered at X is strictly less than 1. This clearlyimplies that the probability that the origin belongs to some ( i )-triangular cover of adjacent( i )-bad squares is strictly less than 1.Given α , β , γ and δ satisfying (5), let ∆ be large enough to satisfy all conditionsimposed on it at the beginning of the proof of Theorem 6. Since in the proof of Theorem6 we take p = (∆ ) − δ − , this implies an appropriate condition on p .The probability that there exists i > X of adjacent ( i )-bad squares suchthat the site (0 ,
0) belongs to the 2 c ( S )∆ i × c ( S )∆ i square centered at X can be boundedfrom above by the expected number of such squares, which is at most X i > (2 c ( S ) + 2) q i c ( S )) X i > ∆ − δi c ( S )) X i > ∆ − δα i . We have δ = α +2 β − − α > p = (∆ ) − δ − > ∆ − δ . Therefore we obtain P p ([ A ] = Z ) c ( S )) X i > p α i / c ( S )) p / + X i > p ( α log αi + α (1 − log α )) / ! , where in the second inequality we use the convexity of the function f ( x ) = α x , whichimplies f ( x ) > f (1) + f ′ (1)( x − p < − / ( α log α ) it follows that P p ([ A ] = Z ) c ( S )) (cid:16) p / + 2 p α/ (cid:17) . (9)Thus if 5( c ( S )) (cid:0) p / + 2 p α/ (cid:1) < p p c ( Z , U ) and the proof of the inequality p c ( Z , U ) > p c ( Z , U ) = 1 if and only if S = S . To show that S 6 = S implies p c ( Z , U ) < Z independently with probability p < Z is not only finite, but isalso surrounded by an annulus of initially infected sites of thickness at least ∇ ( U ). Then,if u ∈ S \ S , we must have an X i ∈ U such that X i ⊂ H u and every finite cluster ofhealthy sites is infected by the dynamics with the use of update rule X i . To show the converse we use following simple argument. Assume that S = S , so thatall update rules in U do not destabilize any direction, i.e., for all i ∈ [ m ] the origin belongsto the convex hull of X i . For any r > p <
1, if we initially infect all sites in Z withprobability p then almost surely somewhere in Z we obtain an initially healthy disk D r of radius r . If r is large enough then every rule X i can only infect sites in disjoint circularsegments “cut off” from D r using chords of length at most ∇ ( U ) and parallel to the sidesof the convex hull of X i , and these segments are all either disjoint or contained in eachother for different rules (that again follows from the fact that we take r large, see Figure8). Because no additional infection takes place in D r , we do not have percolation. Thatcompletes the proof of Theorem 1. Figure 8.
Set of disjoint circular segments cut off from D r using chordsperpendicular to directions u ( θ ) for θ ∈ { π/ , π/ , π/ } . (cid:3) We finally prove the lower bound on p c ( ~ T ,
2) in Corollary 3. We emphasize that becauseour proof is very general, the bounds it gives in specific cases are likely to be far fromoptimal.
Proof of the lower bound in Corollary 3.
For the update family U equivalent to DTBPwe have ∇ ( U ) = dist(( − , − , (0 , √ < .
24. Since in (2) we are free to takeany u , u and u that satisfy this equation for some positive values of the λ i and lieinside open intervals of stable directions that do not intersect the forbidden set, we choose θ ( u ) = 7 π/ θ ( u ) = 23 π/
24 and θ ( u ) = 39 π/
24. This implies that θ ( u ) − θ ( u ) = θ ( u ) − θ ( u ) = θ ( u ) − θ ( u ) = 2 π/
3. Also, for t = 1 , , | θ ( u ) − θ ( u t ) | < π/ u is stable and not forbidden.From these values of θ ( u t ) we get ε > . π and r ε < .
04. This gives c ( S ) = 361.We choose α = 1 . P p ([ A ] = Z ) c ( S )) p / , which isless than 1 when p < − . This implies the condition ∆ > / ( δ +2) . Taking β = 1 . γ = 0 .
01, this condition and the ones at the beginning of the proof of Theorem 6 aresatisfied for ∆ > . Since we have δ = 5 . p c ( ~ T , > . · − andthe proof of Corollary 3 is complete. (cid:3) UBCRITICAL U -BOOTSTRAP PERCOLATION 23 Update families with two opposite strongly stable directions
In this section we present an elementary proof of the fact that the critical probabilityis strictly positive for all update families with two opposite strongly stable directions, i.e.,for families U such that for some u ∈ S we have u, − u ∈ Int S ( U ). The following theoremis of course only a particular subcase of Theorem 1 but it covers all previously analysedsubcritical bootstrap percolation models [24, 27, 20, 21]. (Of course, the point of thosepapers was not, as here, to prove that the critical probability is positive, but rather todetermine quite precise information about its location.) Theorem 9.
For every update family U such that { u, − u } ⊂ Int S ( U ) for some u ∈ S ,we have p c ( Z , U ) > .Proof. Choose u ′ ∈ S and ε > N ε ( u ′ ) , N ε ( − u ′ ) ⊂ Int
S \ F ( U ). Tile Z withidentical rhombi, whose sides are perpendicular to the four directions u ( θ ( ± u ′ ) ± ε/ r > ∇ ( U ). If p > u ′ , ε and r , remains healthy forever. (cid:3) Open problems
When p > p c , the sorts of questions one typically asks of critical bootstrap and U -bootstrap percolation become relevant to subcritical U -bootstrap percolation. For ex-ample, one would like to know about the distribution of the occupation time T of theorigin, and in particular, to what extent this time is concentrated, and how its expecta-tion behaves as p ց p c . These questions have been extensively studied in the case of the r -neighbour model on Z d and are the subject of a number of recent results for criticalupdate families in U -bootstrap percolation. It is natural to ask whether similar behaviouroccurs in the subcritical setting. Some of the following questions (e.g., Question 10 and11) have already been addressed in [27] for models that can be coupled with oriented sitepercolation. However, the methods used in [27] strongly depend on the coupling idea andcannot be applied to “typical” subcritical update families. It is therefore unclear whetherthe models with no two opposite strongly stable directions share similar behaviour. Question 10. (Scaling limit of T .) What is the behaviour of T as p ց p c ? In particular,does T tend to infinity, and if so, what is the limiting dependence of T on p − p c ? The non-triviality of the critical probabilities of subcritical U -bootstrap percolationmodels also opens up the area to the sorts of questions one typically asks of traditionalBernoulli (site or bond) percolation. The difficulty of answering these questions is likelyto be correlated with the difficulty of answering the corresponding questions in Bernoullipercolation: for example, determining the exact value of p c , or even obtaining good boundson p c , for any non-trivial subcritical update family, is likely to be a hard problem. Similarly,properties conjectured to have critical exponent behaviour in Bernoulli percolation, suchas the distribution of cluster sizes, are likely to be hard to analyse in the subcritical U -bootstrap percolation setting. However, there are many properties of site and bond percolation that are now well-understood, at least in two dimensions, and these may alsobe accessible in the subcritical U -bootstrap percolation setting. We give three examples:the behaviour at criticality, exponential decay of cluster sizes, and noise sensitivity. Question 11. (Behaviour at criticality.)
Is there percolation almost surely when p = p c ? If so, do we have E T < ∞ ? Let P p (0 ↔ r ) denote the probability that the origin is contained in a connected com-ponent of radius at least r (according to an arbitrary norm) in the closure of A . Question 12. (Exponential decay.)
For p < p c , does P p (0 ↔ r ) decay exponentiallyin r ? Here we mean ‘connected’ in the site percolation sense, although other notions of con-nectedness are also interesting. It is not clear that one should expect a positive answer toQuestion 12: the droplet-like geometry of the closure of a random initial set suggests thatperhaps the distribution may be much flatter.In the context of random discrete structures, roughly speaking noise sensitivity mea-sures whether small perturbations of a system asymptotically cause all information to belost. The theory of noise sensitivity was introduced by Benjamini, Kalai and Schramm [5],who were motivated by applications to exceptional times in dynamical percolation, and itwas later developed by Garban, Pete, and Schramm [14], and by Schramm and Steif [26].Rather than giving the precise definitions we refer the reader to the articles above for anoverview, and we mention that in the subcritical U -bootstrap percolation setting one candefine a corresponding notion. Question 13. (Noise sensitivity.)
Are subcritical U -bootstrap percolation models noisesensitive at p = p c ? We end with a number of questions of a different flavour, which cannot be asked ofcritical U -bootstrap percolation or of Bernoulli percolation, but which are interesting intheir own right. First, let C ∞ denote the event that there exists an infinite connectedcomponent in the closure of A . Observe that C ∞ is translation invariant, so by ergodicityit has probability either 0 or 1. Combining this with monotonicity, it follows that there isa critical probability p ∞ c = p ∞ c ( U ) such that P p ( C ∞ ) = ( p < p ∞ c p > p ∞ c . It is natural ask about the relationship between p c and p ∞ c : trivially the inequality p ∞ c p c always holds, but is it possible to have strict inequality? Even if not, could it be that P p c ( C ∞ ) = 1 but P p c ([ A ] = Z ) = 0? Question 14. (Infinite component without percolation.)
For which subcritical U -bootstrap percolation models do we have p ∞ c = p c ? This question does not seem to have been studied even in the case of oriented sitepercolation.Define the random variable D ( n ) = (cid:12)(cid:12) [ − n, n ] ∩ [ A ] (cid:12)(cid:12)(cid:12)(cid:12) [ − n, n ] (cid:12)(cid:12) . UBCRITICAL U -BOOTSTRAP PERCOLATION 25 Thus, D ( n ) is the density of the closure [ A ] inside the square [ − n, n ] . Analogous tonumerous phenomena, we conjecture the following. Conjecture 15. (Density of the closure.)
For every p ∈ [0 , there exists a constant δ ( p ) such that D ( n ) converge in probability to a constant δ ( p ) as n → ∞ . This conjecture is one formulation of the assertion that sites in the closure of A shouldbe reasonably well scattered. If Conjecture 15 is true, one would like to know if δ ( p ) iscontinuous at p = p c , and whether we have δ ( p ) − p = o ( p ) as p → U -bootstrap percolation in higher di-mensions. Let d > U be a d -dimensional update family. We definethe stable set in d dimensions completely analogously to in 2 dimensions. First, given( d − S d − ⊂ R d , for each u ∈ S d − , let H du := { x ∈ Z d : h x, u i < } be ahalf-space normal to u . Then the stable set is S = S ( U ) = (cid:8) u ∈ S d − : [ H du ] = H du (cid:9) . Let µ : L ( S d − ) → R be the Lebesgue measure on the collection of Lebesgue-measurablesubsets of S d − . We define the d -dimensional family U to be subcritical if µ ( H ∩ S ) > H ⊂ S d − . Note that this corresponds to the definition given at thestart of the paper in the special case d = 2. We conjecture the following. Conjecture 16.
Fix an integer d > and let U be a d -dimensional update family. Then p c ( Z d , U ) > if and only if U is subcritical. We believe that Conjecture 16 should follow from similar methods to those used in thepresent paper, but with significant technical complications.Our final question concerns directed triangular bootstrap percolation, which was theexample subcritical U -bootstrap percolation process given in the introduction. The lowerbound in Corollary 3 obtained by analysing our proof is likely to be far from the truth.What is the correct value of p c ( ~ T , Question 17.
Can one obtain better bounds on the critical probability p c ( ~ T , for DTBPthan those given in Corollary 3? Finally we remark that there are many other interesting questions that one could andshould ask about subcritical U -bootstrap percolation – too many to list here individually. References
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E-mail address : [email protected] Department of Pure Mathematics and Mathematical Statistics, University of Cambridge,Wilberforce Road, Cambridge CB3 0WB, UK, and Department of Mathematical Sciences,University of Memphis, Memphis, Tennessee 38152, USA, and London Institute for Mathe-matical Sciences, 35a South St, Mayfair, London W1K 2XF, UK
E-mail address : [email protected] Department of Pure Mathematics and Mathematical Statistics, University of Cambridge,Wilberforce Road, Cambridge CB3 0WB, UK, and London Institute for Mathematical Sci-ences, 35a South St, Mayfair, London W1K 2XF, UK
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