Self-organisation in cellular automata with coalescent particles: qualitative and quantitative approaches
SSELF-ORGANISATION IN CELLULAR AUTOMATA WITH COALESCENTPARTICLES: QUALITATIVE AND QUANTITATIVE APPROACHES
BENJAMIN HELLOUIN DE MENIBUS AND MATHIEU SABLIK
Abstract.
This article introduces new tools to study self-organisation in a family of simple cellularautomata which contain some particle-like objects with good collision properties (coalescence) intheir time evolution. We draw an initial configuration at random according to some initial shift-ergodic measure, and use the limit measure to describe the asymptotic behaviour of the automata.We first take a qualitative approach, i.e. we obtain information on the limit measure(s). We provethat only particles moving in one particular direction can persist asymptotically. This provides somepreviously unknown information on the limit measures of various deterministic and probabilisticcellular automata: and -cyclic cellular automata (introduced by Fisch, 1990), one-sided captivecellular automata (introduced by Theyssier, 2004), the majority-traffic cellular automaton, a selfstabilisation process towards a discrete line (introduced by Regnault and Remila, 2015). . .In a second time we restrict our study to to a subclass, the gliders cellular automata. Forthis class we show quantitative results, consisting in the asymptotic law of some parameters: theentry times (generalising [KFD11]), the density of particles and the rate of convergence to the limitmeasure. Introduction
A cellular automaton is a complex system defined by a local rule which acts synchronously anduniformly on a configuration space. These simple models exhibit a wide variety of dynamicalbehaviour and even in the one-dimensional case (the focus of this article) they are not completelyunderstood. Formally, given a finite alphabet A , a configuration is an element of the set A Z . Thisset is compact for the product topology. A cellular automaton F : A Z → A Z is defined by a localfunction f : A [ − r,r ] → A , for some radius r > , which acts synchronously and uniformly on everycell of the configuration: F ( x ) i = f ( x [ i − r,i + r ] ) for all x ∈ A Z and i ∈ Z . Equivalently, cellular automata can be defined as continuous functions that commute with the shiftmap σ defined by σ ( x ) i = x i +1 for all x ∈ A Z and i ∈ Z .Even though cellular automata have been introduced by J. Von Neumann [vN56], the impulsionfor their systematic study was given by the work of S. Wolfram [Wol84]. He instigated a systematicstudy of elementary cellular automata, which are the cellular automata defined on the alphabet { , } with radius (there are = 256 such cellular automata; to each of them we associate anumber n ). In particular he proposed a classification according to the observation of the space-timediagrams produced by the time evolution of cellular automata starting from a random configuration.One of these classes corresponds to a particular form of self-organisation: from a random configu-ration, after a short transitional regime, regions consisting in a simple repeated pattern emerge andgrow in size, while the boundaries between them persist under the action of the cellular automatonand can be followed from an instant to the next. Therefore their movement (time evolution) can bedefined inductively, and in this case we call these boundaries particles . In the simplest case, theseparticles evolve at constant speed and are annihilated when colliding with other particles; however, Key words and phrases.
Cellular automata, Particles, limit measures, Brownian motion. a r X i v : . [ m a t h . D S ] J un hey can sometimes exhibit a periodic behaviour or even perform a random walk, and the collisionsmay give birth to new particles following some more or less complicated rules.This type of behaviour was first observed empirically in elementary cellular automata • identifying and describing the particles, usually as finite words; • describing the particle dynamics and understanding its effect on the properties of the CA.Historically, this study was often performed on individual or small groups of similar-looking CA,and the first step was done in a case-by-case manner. See for example [Fis90b, Fis90a] for the 3-statecyclic automaton, [BF95, BF05] for Rule and other automata with similar dynamics, [Gra84,EN92] for Rule . . . Other works such as [Elo94] skip the first step and study particle dynamicsin an abstract manner, deducing dynamical properties of automata by making assumptions on thedynamics of their particles. This approach was used successfully on probabilistic cellular automatasimulating traffic jams, generalising standard stochastic processes such as the TASEP [GG01].The first general formalism of particles in cellular automata was introduced by M. Pivato: homo-geneous regions correspond to words from a subshift Σ and particles are defects in a configurationof Σ . He developed some invariants to characterise the persistence of a defect [Piv07a, Piv07c] andhe described the different possible dynamics of propagation of a defect [Piv07b].In the present work we focus on the second step first. More precisely, we are interested in howthe existence of some particle set with good dynamical properties affects the typical asymptotic be-haviour. Then we apply this general framework to a variety of examples, finding the sets of particlesby Pivato’s methods or otherwise, to explain the self-organisation that is observed experimentally.Let us define more formally what we mean by typical asymptotic behaviour. Starting from a σ -invariant measure µ ∈ M σ ( A Z ) (i.e. µ ( σ − ( U )) = µ ( U ) for all Borel set U ), we consider theiteration of a cellular automaton F on this measure: F ∗ : M σ ( A Z ) −→ M σ ( A Z ) µ (cid:55)−→ F ∗ µ where F ∗ µ ( U ) = µ ( F − ( U )) for all Borel U. We then study the asymptotic properties of the sequence ( F t ∗ µ ) t ∈ N , and particularly the set ofcluster points called the µ -limit measures set . Sometimes, we are only able to provide informationon the µ -limit set, introduced in [KM00], which is the union of the supports of all limit measures.Equivalently, it is the set of configurations containing only patterns whose probability to appear inthe space-time diagram does not tend to zero as time tends to infinity.When studying typical asymptotic behaviour in this sense, it is unreasonable to expect a generalresult since a wide variety of limit measures can be reached in the general case [HdMS13] and anynontrivial property of the µ -limit set is undecidable [Del11]. That is why we consider restricted casesfor the dynamics of the particles. To determine the µ -limit set in some cases, P. Kůrka suggestsan approach based on particle weight function which assigns weights to certain words [Kůr03].However, this method does not cover any case when a defect can remain in the µ -limit set. Hencewe aim at a more general approach, in terms of particle dynamics as well as initial measures.One of our main motivations for this study is the class of captive cellular automata, where thelocal rule cannot make a colour appear if it is not already present in the neighbourhood. Theseautomata were introduced by G. Theyssier in [The04] for their algebraic properties, but he alsonoticed an interesting phenomenon: when drawing a captive cellular automaton at random (fixedalphabet and neighbourhood), most captive automata exhibited the type of self-organised behaviour escribed above. Any kind of general result regarding self-organisation of captive cellular automataremains a challenging open problem.This article is divided into two main sections, corresponding to improved versions of resultspreviously published in conferences [HdMS11, HdMS12]. In Section 2, we present a qualitativeresult generalised from [HdMS11] with an improved formalism, shorter proofs and a new applicationto probabilistic cellular automata (Section 2.7). Then, in Section 3, we refine our approach on asubclass to obtain some quantitative results. Sections 3.1 to 3.3 were published in [HdMS12]; wecorrect some inaccuracies in the proofs and extend the study to other parameters. Qualitative approach.
In Section 2, we prove a qualitative result: for any initial σ -ergodic measure µ , assuming particles have good collision properties (coalescence), only particles moving in oneparticular direction can persist aymptotically. We introduce our own formalism of particle systemin Section 2.1 so as to be able to describe the dynamics of the particles, and Section 2.4 is dedicatedto the proof itself. Section 2.5 presents a simplified version of Pivato’s formalism which is by farthe simplest way to find such a particle system in most examples.We spend Section 2.6 on various examples of automata where this result can be applied: Section 2.6.1: we characterise the µ -limit set of the “traffic” automaton (rule ), a sim-ple case that may clarify the formalism. The results were known for initial Bernoulli mea-sures [BF95, BF05] but our method applies for every σ -ergodic measure. Section 2.6.2: we consider the family of n -cyclic cellular automata introduced in [Fis90a,Fis90b]. Using our method, we go further in the study of these simple automata: in partic-ular, for n = 3 or , we show that the limit measure is unique and is a convex combinationof Dirac measures supported by uniform configurations. Section 2.6.3: we characterise the µ -limit set of all one-sided captive cellular automata. Thisis a first step to the study of asymptotic behaviour of captive cellular automata. Section 2.6.4: last, we apply our formalism to a cellular automaton where the particles donot have a linear speed but instead perform random walks by drawing randomness from theinitial measure.However, our results are not general enough to apply to defects of a sofic subshift that can have aparticle-like behaviour, such as in Rule (see the bottom right picture in Figure 1 and [EN92]),or to more complicated particle systems such as those observed in general captive cellular automata.Finally, in Section 2.7, we generalise our method to probabilistic cellular automata. As an appli-cation, we partially describe limit measures of the probabilistic majority-traffic cellular automatonproposed by N. Fatès in [Fat13] as a candidate to solve the density classification problem. Thiscomplements the approach of [BFMM13] which characterises invariant measures. Another appli-cation proposed in Section 2.7.3 presents a generalisation to the infinite line of a self stabilisationprocess toward a discrete line proposed in [RR15].
Quantitative approach.
In Section 3, we improve the previous qualitative results with a quanti-tative approach, considering the time evolution of some parameters when the particle dynamics arevery simple. This research direction was inspired by [KFD11], where the authors consider the wait-ing time before a particle crosses the central column (called entry time). Using the same approachas in [BF95, KM00], we show that the behaviour of these automata can be described by a randomwalk process (Section 3.1), and we approximate this process by a Brownian motion using scale in-variance (Section 3.3). Thanks to this tool, we answer negatively a conjecture proposed in [KFD11]by determining the correct asymptotic law for the entry time of a particle in the central column(Section 3.2). We then use the same approach on various natural parameters such as the density ofparticles at time t (Section 3.4) or the rate of convergence to the limit measure (Section 3.5). Thisgeneralises some known results on initial Bernoulli measures from [KFD11] and [BF05], in particular elaxing the conditions on the initial measures. In Section 3.6, we exhibit various examples withsimilar dynamics on which these results apply.In all the article, space-time diagrams were produced using the Sage mathematical software [S + (cid:3) = 0 , (cid:4) = 1 , (cid:4) = 2 , (cid:4) = 3 . C a s e s w i t h q u a n t i t a t i v e r e s u l t s ( S ec t i o n ) (-1,1)-gliders CA (Sec. 3.1) Rule 184 or traffic rule (Sec. 2.6.1) 3-state cyclic CA (Sec. 2.6.2) C a s e s w i t h q u a li t a t i v e r e s u l t( S ec t i o n ) (0,1)-gliders CA (Sec. 3.1) One-sided captive CA (Sec. 2.6.3) One-sided captive CA (Sec. 2.6.3)4-state cyclic CA (Sec. 2.6.2) 5-state cyclic CA (Sec. 2.6.2) Random walk CA (Sec. 2.6.4) P r o b . C A ( S ec . . ) Majority-traffic PCA (Sec. 2.7.2) Self-stabilisation of the line (Sec. 2.7.3) Generic one-sided captive PCAGeneric captive CA Generic captive CA Generic captive CA U n k n o w n c a s e s Generic captive CA Generic captive CA Rule 18
Figure 1.
Space-time diagrams of some cellular automata with particles, startingfrom a configuration drawn uniformly at random. . Particle-based organisation: qualitative results
In this section, we take a qualitative approach to self-organisation: that is, we assume some prop-erties on the dynamics of the particles of some cellular automaton and try to deduce properties ofits µ -limit measures set, with no regard to how fast this organisation takes place. Particles
Definition of symbolic systems
Given a finite alphabet A , a word is a finite sequence of elements of A . Denote by A ∗ = (cid:83) n A n the set of all words where A is the empty word ε . An infinite sequence indexed by Z is calleda configuration . The set of configurations A Z is a compact set for the product topology. For aword u ∈ A ∗ the cylinder [ u ] is the set of configurations where u appears at the position , and for U ⊂ A ∗ we have [ U ] = (cid:83) u ∈ U [ u ] . Cylinders are a clopen basis of the topology.On A Z we define the shift map σ ( x ) i = x i +1 for all x ∈ A Z and i ∈ Z . A subshift is a closed σ -invariant subset of A Z . Equivalently, a subshift can be defined by a set of forbidden patterns F ⊂ A ∗ as the set of configurations where no pattern of F appears. If F is finite, we call thecorresponding subshift a subshift of finite type or SFT. The radius of an SFT Σ is the smallest (cid:96) such that Σ can be defined by a set of forbidden patterns in A (cid:96) . The language of a subshift Σ isdefined as L n (Σ) = { u ∈ A n : Σ ∩ [ u ] (cid:54) = ∅} and L (Σ) = (cid:83) n ∈ N L n (Σ) . A SFT is σ -transitive if forany two patterns u, v ∈ L (Σ) , there exists w ∈ A ∗ such that uwv ∈ L (Σ) .Given two finite alphabets A and B , a morphism from A Z to B Z is a continuous function π : A Z → B Z which commutes with the shift (i.e. σ ( π ( x )) = π ( σ ( x )) for all x ∈ A Z ). Equivalentlya morphism can be defined by a local map f : A N → B where N ⊂ Z is a finite set called theneighbourhood such that π ( x ) i = f ( x i + N ) for all x ∈ A Z and i ∈ Z .The radius of π is the minimal r ∈ N such that π admits a local map with N ⊂ [ − r, r ] . A cellularautomaton is a morphism from A Z to itself, that is, the input and the output are defined on thesame alphabet. In particular a cellular automaton can be iterated and it makes sense to study itsdynamics.2.1.2. Particle systemDefinition 1 (Particle system) . Let F : A Z → A Z be a cellular automaton. A particle system for F is a triplet ( P , π, φ ) , where: • P is a finite set of elements called particles ; • π : A Z (cid:55)→ ( P ∪{ } ) Z is a morphism identifying the presence of particles at each position; Theset of positions that carry particles on x is denoted by Part P ,π ( x ) = { k ∈ Z : π ( x ) k ∈ P} (we omit P and π when they are clear from the context); • φ : A Z × Z (cid:55)→ Z (where Z denotes the set of subsets of Z ) is a function called the updatefunction that describes the movement and/or offsprings of each particle after one iterationof F ;such that the following conditions are satisfied for all x ∈ A Z and k ∈ Z : Locality:
There is a constant r > (the radius of the system) such that φ ( x, k ) ⊂ [ k − r, k + r ] .The particles cannot “jump” arbitrarily far; the radius does not depend on x and k . Redistribution: (cid:83) k ∈ Part ( x ) φ ( x, k ) = Part ( F ( x )) (cid:83) k / ∈ Part ( x ) φ ( x, k ) = ∅ .The particle in F ( x ) are exactly the offsprings of particles of x , and non-particles do nothave offsprings. isjunction: k < k (cid:48) ⇒ φ ( x, k ) = φ ( x, k (cid:48) ) or max φ ( x, k ) < min φ ( x, k (cid:48) ) .Two particles either do not interact (in which case they cannot cross), or they share thesame set of offsprings.The four conditions ensure that the update function accurately describes the time evolution ofthe particles. Notice that since the morphism and update function are defined locally, the conditionscan be checked algorithmically by simple enumeration of patterns up to a certain length.In the context of a fixed particle system for F , we use shorthands for the composition of theupdate function, defined inductively: φ t ( x, k ) = (cid:91) k (cid:48) ∈ φ ( x,k ) φ t − ( F ( x ) , k (cid:48) ) , and a notion of pre-image (with an abuse of notation): φ − x ( A ) = { k ∈ Z | φ ( x, k ) ∩ A (cid:54) = ∅} . If φ ( x, k ) is a singleton, we use “ φ ( x, k ) ” instead of “the only member of φ ( x, k ) ” as an abuse ofnotation.2.1.3. Coalescence
We postpone the discussion on how to find a particle system in a given cellular automaton toSection 2.5. We now look for assumptions on the dynamics of the particles that let us deduce thatsome particles disappear asymptotically. Simulations suggest that this is the case when the particlesare forced to collide, and that these collisions are destructive in the sense that the total number ofparticles decreases; thus we introduce the notion of coalescence.
Definition 2 (Coalescence) . Let F : A Z → A Z be a cellular automaton, and ( P , π, φ ) a particlesystem for F . This particle system is coalescent if, for every x ∈ A Z and k ∈ Part ( x ) , the particlehas one of the two following behaviours: Progression: | φ ( x, k ) | = | φ − x ( φ ( x, k )) | = 1 , and π ( x ) k = π ( F ( x )) φ ( x,k ) (the particle persists and its type does not change), or Destructive interaction: | φ ( x, k ) | < | φ − x ( φ ( x, k )) | (particles collide and generate strictly fewer particles (possibly 0); or a single particledisappears).Progressing and interacting particles of a configuration x ∈ A Z are denoted by Prog P ,π,φ ( x ) andInter P ,π,φ ( x ) , respectively, and P , π and φ are omitted when the particle system is clear from thecontext. k ∈ Prog P ,π,φ ( x ) is the case when we use “ φ ( x, k ) ” to mean “the only member of thesingleton φ ( x, k ) ”. Probability measures and µ -limit sets The µ -limit set was introduced in [KM00] to describe the asymptotic behaviour corresponding toempirical observations. It consists in the patterns whose probability to appear does not tend to 0when the initial point is chosen at random. To define it formally, let us introduce some notations.Denote by M σ ( A Z ) the set of σ -invariant probability measures on A Z (i.e. measures µ such that µ ( σ − ( U )) = µ ( U ) for any Borel set U ). A measure is σ -ergodic if every σ -invariant Borel set hasmeasure or , and we denote by M σ − erg ( A Z ) the set of σ -ergodic probability measures. [Wal82]gives a good introduction to ergodic probability measures. xamples. The
Bernoulli measure λ ( p a ) a ∈A associated with a sequence ( p a ) a ∈A of elements of [0 , whose sum is is defined by λ ( p a ) a ∈A ([ u ]) = p u p u . . . p u | u |− for all u ∈ A ∗ . If all the elements of ( p a ) a ∈A have the same value | A | we call it the uniform Bernoulli measure and denote it by λ . Forany finite word u ∈ A ∗ , define (cid:98) δ u as the unique σ -invariant probability measure supported by the σ -periodic configuration ω u ω and its translations.Given a cellular automaton F : A Z → A Z and an initial measure µ ∈ M σ ( A Z ) , we define themeasure F ∗ µ by F ∗ µ ( U ) = µ ( F − ( U )) for any Borel set U . Since F commutes with σ , one has F ∗ µ ∈ M σ ( A Z ) . Moreover if µ ∈ M σ − erg ( A Z ) , then F ∗ µ ∈ M σ − erg ( A Z ) as well. This allows todefine the following action: F ∗ : M σ ( A Z ) −→ M σ ( A Z ) µ (cid:55)−→ F ∗ µ. We consider the set of cluster points of the sequence ( F t ∗ µ ) t ∈ N called the µ -limit measures set anddenoted by V ( F, µ ) . The closure of the union of the supports of these measures is called the µ -limitset and it is denoted by Λ µ ( F ) . Equivalently, it can be defined as the subshift Λ µ ( F ) = (cid:110) x ∈ A Z : ∀ i, j ∈ Z , F t ∗ µ ([ x [ i,j ] ]) (cid:57) (cid:111) . See [KM00] for all basic examples. The µ -limit (measures) set has been also well studied for twoclasses of cellular automata : automata exhibiting particle-like behaviour ([Fis90a, BF05] and manyothers) and automata with an algebraic structure ([Lin84] and others). Evoution of the density of particles for coalescent systems
Define the frequency with which the pattern u appears in the configuration x asFreq ( u, x ) = lim sup n →∞ Card { i ∈ [ − n, n ] : x [ i,i + | u |− = u } n + 1 . Similarly we define Freq ( S, x ) where S is a set of patterns.We introduce the following notations for all the subsequent proofs. For n ∈ N , let B n be the set [ − n, n ] ⊂ Z . Let F : A Z → A Z be a cellular automaton. In the context of a fixed particle system ( P , π, φ ) , the densities of particles in a configuration x ∈ A Z are defined by:for p ∈ P , D p ( x ) = Freq ( p, π ( x )) and D ( x ) = Freq ( P , π ( x )); D Prog ( x ) = lim sup t →∞ t + 1 | Prog ( x ) ∩ B t | and similarly for D Inter ( x ) , the last two definitions applying only if the particle system is coalescent.For µ ∈ M σ − erg ( A Z ) , by Birkhoff’s ergodic theorem, the lim sup can be replaced by a simplelimit in the definition of frequency for µ -almost all configurations. This implies for example that D ( x ) = (cid:80) p ∈P D p ( x ) for µ -almost all x .First of all, the following proposition clarifies how controlling the frequency of interactions givesus information about the evolution of the density of the different kinds of particles. Proposition 1 (Evolution of densities) . Let F : A Z → A Z be a cellular automaton, µ ∈ M σ − erg ( A Z ) , ( P , π, φ ) a coalescent particle system for F , and r the radius of the update function φ . Then, for µ -almost all x ∈ A Z : (1) D ( F ( x )) ≤ D ( x ) − r +1 D Inter ( x ) ; (2) ∀ p ∈ P , D p ( F ( x )) ≤ D p ( x ) + D Inter ( x ) . Lemma 1. (1)
For all x and k : | φ ( x, k ) | + | φ − x ( φ ( x, k )) | ≤ r + 2 , which implies when k ∈ Inter ( x ) : | φ ( x, k ) | ≤ rr + 1 | φ − x ( φ ( x, k )) | . Proof. (of Lemma 1) Take i ≤ i (cid:48) , resp. j ≤ j (cid:48) , the extremal points of | φ ( x, k ) | and | φ − x ( φ ( x, k )) | respectively. By locality of the update function, we have: | φ ( x, k ) | + | φ − x ( φ ( x, k )) | ≤ ( i (cid:48) − i + 1) + ( j (cid:48) − j + 1) = ( i (cid:48) − j ) + ( j (cid:48) − i ) + 2 ≤ r + 2 . The proof is illustrated in Figure 2. xF ( x ) φ − x ( φ ( x, k )) φ ( x, k ) ≤ r ≤ r Figure 2.
Visual proof that | φ ( x, k ) | + | φ − x ( φ ( x, k )) | ≤ r + 2 .If furthermore k ∈ Inter ( x ) , since the particle system is coalescent, we have | φ ( x, k ) | < | φ − x ( φ ( x, k )) | .The maximum of the ratio | φ ( x,k ) || φ − x ( φ ( x,k )) | is then reached on | φ ( x, k ) | = r , | φ − x ( φ ( x, k )) | = r + 1 . (cid:3) We continue the proof of Proposition 1
Proof. (1) By the redistribution property of the update function, we have Part ( F ( x )) = (cid:83) k ∈ Part ( x ) φ ( x, k ) .Furthermore, by locality, ∀ x ∈ A Z , ∀ n ∈ N , Part ( F ( x )) ∩ B n ⊆ (cid:91) k ∈ Part ( x ) ∩ B n + r φ ( x, k ) ⊆ (cid:91) k ∈ Prog ( x ) ∩ B n + r φ ( x, k ) (cid:116) (cid:91) k ∈ Inter ( x ) ∩ B n + r φ ( x, k ) , where (cid:116) denotes a disjoint union. The second line is obtained by coalescence: since Part ( x ) = Prog ( x ) (cid:116) Inter ( x ) , particles in F ( x ) are either images of progressing particles or of interactingparticles. By disjunction: ∀ x ∈ A Z , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:91) k ∈ Prog ( x ) ∩ B n + r φ ( x, k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = | Prog ( x ) ∩ B n + r | and ∀ x ∈ A Z , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:91) k ∈ Inter ( x ) ∩ B n + r φ ( x, k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ rr + 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) φ − x (cid:91) k ∈ Inter ( x ) ∩ B n + r φ ( x, k ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ rr + 1 | Inter ( x ) ∩ B n +2 r | . This first equality is because progressing particles are “one-to-one”. The second inequality is byLemma 1 and by locality. It follows: ∀ x ∈ A Z , | Part ( F ( x )) ∩ B n | ≤ | Prog ( x ) ∩ B n + r | + rr + 1 | Inter ( x ) ∩ B n +2 r | . Then, passing to the limit:For µ -almost all x ∈ A Z , D ( F ( x )) ≤ D Prog ( x ) + rr + 1 D Inter ( x ) = D ( x ) − r + 1 D Inter ( x ) .
2) Similarly, for any particle p ∈ P , one has for all x ∈ A Z and n ∈ N : { k ∈ B n | π ( F ( x )) k = p } ⊆ (cid:91) k ∈ Part ( x ) ∩ B n + r φ ( x, k ) (locality) . For k ∈ Prog ( x ) , if π ( F ( x )) φ ( x,k ) = p , then by definition of coalescence π ( x ) k = p . For µ -almost all x , using Part ( x ) = Prog ( x ) (cid:116) Inter ( x ) , we conclude that D p ( F ( x )) ≤ D p ( x ) + D inter ( x ) by passingto the limit. (cid:3) A particle-based self-organisation result
We state our main result. A simple version (Corollary 1) states that in a coalescent particle systemwith a σ -ergodic initial measure, if all particles can be assigned a speed, then only particles withone fixed speed may survive asymptotically. The more general result is designed to handle moredifficult cases such as particles performing random walks, as in the last example of Section 2.6. Definition 3 (Clashing) . Let F : A Z → A Z be a cellular automaton, ( P , π, φ ) a coalescent particlesystem for F , and P and P two subsets of P . We say that P clashes with P µ -almost surely if,for every n ∈ N ∗ and µ -almost all x ∈ A Z , π ( x ) ∈ P and π ( x ) n ∈ P = ⇒ ∃ t ∈ N , φ t ( x, ∈ Inter ( F t ( x )) or φ t ( x, n ) ∈ Inter ( F t ( x )) The abuse of notation in the last line is justified by the fact that, if the images φ t ( x, k ) ( k = 0 , n )are not in interaction for all t (cid:48) < t , then φ t ( x, k ) is still a singleton.The intuition behind clashing particles is the following: if two clashing particles are presentwith positive frequency, then at least one of them end up almost surely in interaction with positivefrequency, decreasing the global frequency of particles. This is why they cannot both persist asymp-totically. Note that clashing is oriented left to right: intuitively, particles with speed +1 clash withparticles with speed − , but the converse is not true. Theorem 1 (Main qualitative result) . Let F : A Z → A Z be a cellular automaton, µ an initial σ -ergodic measure and ( P , π, φ ) a coalescent particle system for F where P can be partitioned intosets P . . . P n such that, for every i < j , P i clashes with P j µ -almost surely.Then: (1) All particles appearing in the µ -limit set belong to the same subset, i.e. ∃ i ∈ [1 , n ] , ∀ p ∈ P , p ∈ L ( π (Λ µ ( F ))) ⇒ p ∈ P i . (2) If furthermore there exists a j such that P j clashes with itself µ -almost surely, then thissubset of particles does not appear in the µ -limit set, i.e. ∀ p ∈ P , p ∈ L ( π (Λ µ ( F ))) ⇒ p / ∈ P j . We introduce the notion of speed which is less general but easier to handle than the notion ofclashing.
Definition 4 (Speed) . Let F be a cellular automaton and ( P , π, φ ) be a particle system for F .A particle p ∈ P has speed v ∈ Z if for any configuration x ∈ A Z and k ∈ Z such that π ( x ) k = p ,we have one of the following: Eventual interaction: ∃ t, φ t ( x, k ) ∈ Inter ( F t ( x )) ; Progression at speed v : ∀ t, φ t ( x, k ) ∈ Prog ( F t ( x )) and φ t ( x,k ) − kt → t →∞ v . Corollary 1 (Version with speedy particles) . Let F : A Z → A Z be a cellular automaton, µ aninitial σ -ergodic measure and ( P , π, φ ) a coalescent particle system for F .If each particle p ∈ P has speed v p ∈ R ,then there is a speed v ∈ R such that: ∀ p ∈ P , p ∈ L ( π (Λ µ ( F ))) ⇒ v p = v. roof of Theorem 1. For the first point, Let i = 1 , j = 2 for clarity and let p ∈ P , p ∈ P be twoparticles. We show that they cannot both appear in the µ -limit set.First we study the behaviour of the sequences of density of particles. For all x ∈ A Z , byProposition 1(1), ( D ( F t ( x ))) t ∈ N is a decreasing sequence of positive reals and admits a limit d ∞ ( x ) . In particular D Inter ( x ) → . Applying Birkhoff’s theorem to π ∗ F t ∗ µ for any t , we getthat D ( F t ( x )) = π ∗ F t ∗ µ ([ P ]) for µ -almost all x (recall that [ P ] = (cid:83) p ∈P [ p ] ). In particular there is areal d ∞ such that d ∞ ( x ) = d ∞ for µ -almost all x .For x ∈ A Z , we define D P i ( x ) = Freq ( P i , π ( x )) ; we prove that this sequence also admits a limit.By Proposition 1(2), we have:For i ∈ { , } , sup n ∈ N |D P i ( F t + n ( x )) − D P i ( F t ( x )) | ≤ ∞ (cid:88) n =0 D Inter ( F t + n ( x )) . To prove that ( D P i ( F t ( x ))) t ∈ N is a Cauchy sequence, we need to show that (cid:80) t ∈ N D Inter ( F t ( x )) < + ∞ . By Proposition 1(1), we have: (cid:88) t ∈ N D Inter ( F t ( x )) ≤ ( r + 1) (cid:32)(cid:88) t ∈ N D ( F t ( x )) − D ( F t +1 ( x )) (cid:33) ≤ ( r + 1)( D ( x ) − d ∞ ( x )) < + ∞ . Thus ( D P i ( F t ( x ))) t ∈ N is a Cauchy sequence and admits a limit d i ( x ) (cid:54) = 0 . Using again Birkhoff’stheorem, we have that ( D P i ( F t ( x ))) t ∈ N = ( π ∗ F t ∗ µ ([ P i ])) t ∈ N for µ -almost all x , and therefore thereis a real d i such that d i ( x ) = d i for µ -almost all x .Assume that p i ∈ L ( π (Λ µ ( F ))) for i = 1 , . This implies d i > for i = 1 , . Since clashingparticles generate interactions, we show that this contradicts the fact that (cid:80) D Inter ( F t ( x )) < + ∞ for all x .Fix ε < d · d r +3 and T large enough such that for t ≥ T, π ∗ F t ∗ µ ([ P ]) − d ∞ < ε and | π ∗ F t ∗ µ ([ P i ]) − d i | <ε for i ∈ { , } . By Birkhoff’s ergodic theorem applied on π ∗ F T ∗ µ , we have: K K (cid:88) k =0 π ∗ F T ∗ µ ([ p ] ∩ [ p ] k ) −→ K →∞ π ∗ F T ∗ µ ([ p ]) · π ∗ F T ∗ µ ([ p ]) . Note that [ p ] ∩ [ p ] k are words containing clashing particles positioned so that they will generatean interaction. We have π ∗ F T ∗ µ ([ p ]) · π ∗ F T ∗ µ ([ p ]) ≥ ( d − ε ) · ( d − ε ) ≥ d · d − ε . ByBirkhoff’s theorem, this means that for µ -almost all x ∈ A Z , words belonging in (cid:83) k V k where V k = p ( P ∪ { } ) k − p ⊂ ( P ∪ { } ) ∗ have frequency at least d · d − ε in πF T ( x ) .Since P and P clash µ -almost surely, any occurrence of V k yields a future interaction: that is,Freq (cid:0)(cid:83) k V k , πF T ( x ) (cid:1) ≤ (cid:80) ∞ t = T D Inter ( F t ( x )) . We show the contradiction:For µ -almost all x ∈ A Z , D ( F T ( x )) − d ∞ ≥ r + 1 ∞ (cid:88) t = T D Inter ( F t ( x )) Proposition 1(i) ≥ r + 1 ( d · d − ε ) > ε, which is a contradiction with the definition of ε . To prove the second point, apply the same proofto two particles in P j . (cid:3) Proof of Corollary 1.
Consider the set of speeds { v p : p ∈ P} and order it as v > v > · · · > v n .Now partition the set of particles into ( P v i ) ≤ i ≤ n where P v i is the set of particles with speed v i ,and apply the Theorem 1.We check the hypothesis of Theorem 1: for any i < j , P v i clashes with P v j µ -almost surely. Let p i ∈ P v i and p j ∈ P v j , and x ∈ A Z such that π ( x ) = p i and π ( x ) n = p j for some n ∈ N ∗ . If oth particles satisfy the second property in the definition of speed (Progression at speed v ), thenfor some t large enough we have φ t ( x, > φ t ( x, n ) , which is forbidden by coalescence since twoparticles in progression cannot cross. Thus at some time t we have either φ t ( x, ∈ Inter ( F t ( x )) or φ t ( x, n ) ∈ Inter ( F t ( x )) . (cid:3) Pivato’s defect formalism
Before giving a series of examples where this result can be used to describe the typical asymptoticbehaviour of a cellular automaton, we present the formalism introduced by Pivato in [Piv07a, Piv07c]that defines particles as defects with respect to a F -invariant subshift Σ . Indeed, this formalismgives us an easier way to find the particle systems in our examples.Intuitively, the F -invariant subshift describes the homogeneous regions that persist under theaction of F in the space-time diagram, and defects are the borders between these regions. Thisallows us to define P and π in a way that corresponds to the intuition, even though it gives noinformation on the dynamics of the particles (the update function φ ).2.5.1. Defects
For a cellular automaton F , consider Σ a F -invariant subshift. The defect field of x ∈ A Z withrespect to Σ is defined as: F Σ x : Z → N ∪ {∞} k (cid:55)→ max (cid:110) n ∈ N : x k +[ −(cid:98) n − (cid:99) , (cid:100) n − (cid:101) ] ∈ L n (Σ) (cid:111) , where the result is possibly or ∞ if the set is empty or infinite. Intuitively, this function returnsthe size of the largest word admissible for Σ centred on the cell k . A defect in a configuration x relative to Σ is a local minimum of F Σ x . Then the interval [ k, l ] between two defects forms ahomogeneous region in the sense that x [ k +1 ,l ] ∈ L (Σ) .However, it is not true that we can always make a correspondence between defects and a finiteset of words (forbidden patterns), so as to obtain a finite set of particles and a morphism. This isthe case only when the set of forbidden patterns is finite, that is, when Σ is a SFT. In this case, adefect corresponds to the centre of the occurrence of a forbidden word. This is a limitation of ourresult.The examples given in Figure 1 suggest that defects can usually be classified using one of theseapproaches: • Regions correspond to different subshifts and defects behave according to their surroundingregions ( interfaces - e.g. cyclic automaton); • Regions correspond to the same periodic subshift and defects correspond to a “phase change”( dislocations - e.g. rule 184 automaton).2.5.2.
Interfaces
Assume that Σ is a SFT and can be decomposed as a disjoint union Σ (cid:116) · · · (cid:116) Σ n of F -invariant σ -transitive SFTs (the domains ). Intuitively, the region between two defects belongs to the languageof (at least) one of the domains; we classify each defect according to which domain the regionssurrounding it on the left and on the right correspond to. Since each domain is F -invariant, thisclassification is conserved under the action of F for non-interacting defects.Formally, since the different domains (Σ k ) k ∈ [1 ,n ] are disjoint SFTs, there is a length α > such that ( L α (Σ k )) k ∈ [1 ,n ] are disjoint (if two subshifts share arbitrarily long words, they share aconfiguration by closure). In particular, if u ∈ L α (Σ) , then there is a unique k such that u ∈ L (Σ k ) :we say that u belongs to the domain k . Thus, for a given configuration, we can assign a choice of domain to each homogeneous region between two consecutive defects, and this choice is unique ifthis region is larger than α cells.
17 15 13 11 02 04 05 03 01 22 24 25 23 21 02 04 06 − − − F a ( x ) x domain. . . . . . Figure 3.
Interfaces between monochromatic domains, marked by slanted patterns.To each interface corresponds a domain change, marked by a red line.Defects relative to such an SFT are called interface defects and can be classified according to thedomain of the surrounding regions. Let P = { p ij : ( i, j ) ∈ [1 , n ] } be the set of particles. Definethe morphism π : A Z → ( P ∪ { } ) Z of radius max( (cid:100) r/ (cid:101) , α ) , where r is the radius of Σ , in thefollowing way. For x ∈ A Z and k ∈ Z : • if x k +[ −(cid:98) r (cid:99) , (cid:100) r (cid:101) ] ∈ L (Σ) , then π ( x ) k = 0 ; • else, let u = x [ k − m,k ] where m = max { (cid:96) ≤ α : x [ k − (cid:96),k ] ∈ L (Σ) } u = x [ k +1 ,k + m ] where m = max { (cid:96) ≤ α : x [ k +1 ,k + (cid:96) ] ∈ L (Σ) } d i a domain to which u i belongs ( i ∈ { , } ) and put π ( x ) k = p d d .The domain choice (choice of d i ) is unique when domains contain at least α cells; otherwise, thechoice between the possible d i is arbitrary, or fixed beforehand. Notice that the first check requiresradius at least (cid:100) r (cid:101) and the second check requires radius at least α .2.5.3. Dislocations
Contrary to interface defects that mark a change between languages of different SFT, dislocationdefects mark a “change of phase” inside a single SFT.Let Σ be a σ -transitive SFT of order r > . We say that Σ is P -periodic if there exists a partition V , . . . , V P of L r − (Σ) such that a · · · a r ∈ L r (Σ) ⇔ ∃ i ∈ Z /P Z , a · · · a r − ∈ V i and a · · · a r ∈ V i +1 . The period of Σ is the maximal P ∈ N such that Σ is P -periodic. For example, the orbit of a finiteword u ∈ A ∗ , defined as { σ k ( ∞ u ∞ ) : k ∈ Z } is a periodic SFT of period at most | u | .We thus associate to each x ∈ Σ its phase ϕ ( x ) ∈ Z /P Z such that x [0 ,r − ∈ V ϕ ( x ) . Obviously, ϕ ( σ k ( x )) = ϕ ( x ) + k mod p . For x ∈ A Z , we say that an homogeneous region [ a, b ] (i.e. a regionsuch that x [ a,b ] ∈ Σ ) is in phase k if ∃ y ∈ Σ , ϕ ( y ) = k, x [ a,b ] = y [ a,b ] . If b − a > r − , the phase of aregion is unique and means x [ a,a + r − ∈ V k + a mod p .
17 15 13 11 02 04 06 05 03 01 12 14 15 13 11 02 04 06 − − − F a ( x ) x phase. . . . . . Figure 4.
Dislocations in the chequerboard subshift ( P = 2 ), marked by slantedpatterns. Red lines show the visual intuition of a change of phase, with the sur-rounding local phases. s we can see in Figure 4, the finite word corresponding to a defect (here or ) does notdepend only on the phase of the surrounding region but also on the position of the defect. Moreprecisely, since ϕ ( σ ( x )) = ϕ ( x ) + 1 , a defect in position j with a region in phase f to its left and adefect in position with a region in phase f + j mod P to its left “observe” the same finite wordto their left.Therefore, we define for each defect its local phases . Assume a defect is in position j surroundedby homogeneous regions [ i, j ] and [ j, k ] in phase ϕ (cid:96) and ϕ r , respectively. Then its left local phase (resp. right local phase ) is ϕ (cid:96) + j mod P , resp. ϕ r + j mod P .Now we classify the defects according to the local phase of the surrounding regions. Let P = { p ij : ( i, j ) ∈ Z /P Z } be the set of particles. Since defects correspond to the centre of occurrencesof forbidden words and the phase of a region can be locally distinguished, the morphism π : A Z → ( P ∪ { } ) of order r − is defined exactly as in the interface case. The choice of local phase isunique if the region is larger than r − cells.In the general case, those two formalisms can be mixed by fixing a decomposition Σ = (cid:70) i ∈A Σ i where some of the Σ i have nonzero periods. We can classify defects according to the domains andlocal phase of the surrounding regions in a similar manner. Except for the arbitrary choices forsmall regions, obtaining the set of particles and the morphism from the SFT decomposition can bedone in an automatic way. Examples
Rule 184
We consider the rule or “traffic” automaton F : { , } Z → { , } Z defined by the followinglocal rule: f ( x − x x ) = 1 if and only if x x = 11 or x − x = 10 .The time evolution of this automaton can be seen as a road where the symbol represent vehiclesand the symbol an empty space. The vehicles move forward if the cell in front of them is emptyand stay put otherwise. In this context, the rule has been very well studied, especially in thecase of initial Bernoulli measures [BF95, BF05]. We use this example mostly as a simple case tobetter understand the formalism, although our method has the advantage to hold for more generalprobability measures. Proposition 2.
Let F be the traffic automaton and µ ∈ M σ − erg . Then: µ ([00]) > µ ([11]) ⇒ / ∈ L (Λ µ ( F )); µ ([00]) < µ ([11]) ⇒ / ∈ L (Λ µ ( F )); µ ([00]) = µ ([11]) ⇒ F t ∗ µ → (cid:99) δ . → Figure 5.
Particle system for the traffic automaton. roof. We consider the chequerboard SFT
Σ = { ∞ (01) ∞ , ∞ (10) ∞ } , which is -periodic and F -invariant. Using the dislocation formalism, we define the phases ϕ ( ∞ (01) ∞ ) = 0 and ϕ ( ∞ (10) ∞ ) =1 , obtaining a set of particles defined by their local phases { p , p } . The corresponding morphismof order r = 2 is defined by the local rule: → p → p otherwise → . Indeed, consider x ∈ A Z with a defect x = 00 . The phase of the in position is and the phaseof the in position is , so this corresponds to a particle p . Changing the position of the defectwould not change the particle since the local phase would be modified accordingly.The update function is defined in the intuitive manner: with p evolving at speed +1 and p at speed − and both particles being sent to ∅ in case of collision. ∀ x ∈ A Z , ∀ k ∈ Z , φ ( x, k ) = { k − } if π ( x ) k = p and π ( x ) k − (cid:54) = p { k + 1 } if π ( x ) k = p and π ( x ) k +2 (cid:54) = p ∅ otherwise (and in particular if π ( x ) k = 0) We now check that the particle system satisfies all necessary conditions. To do that, one shouldverify that the update function is defined properly, that is: ∀ x ∈ A Z , ∀ k ∈ Z , π ( F ( x )) k − = p ⇔ F ( x ) { k − ,k } = 11 ⇔ x [ k − ,k +1] ∈ { , , }⇔ π ( x ) k = p and π ( x ) k +2 (cid:54) = p , and similarly for p . This type of proof can become tedious due to the high number of cases butcan be automated by straightforward enumeration, here of all patterns of length . The differentconditions follow from this property: Locality:
Obvious by definition of φ . Redistribution:
The claim can be restated as π ( F ( x )) k +1 = p ⇔ π ( x ) k = p and φ ( x, k ) = { k +1 } , and similarly for p . The first condition follows. Since φ ( x, k ) = ∅ when k / ∈ Part ( x ) by definition of φ , the second condition follows. Disjunction:
For k < k (cid:48) , to have φ ( x, k ) > φ ( x, k (cid:48) ) , the only way would be to have π ( x ) k (cid:48) = p , π ( x ) k = p and k (cid:48) = k + 1 . In that case, by definition, φ ( x, k ) = φ ( x, k (cid:48) ) = ∅ . Coalescence and speeds:
Obvious by definition of φ .Therefore we can apply Corollary 1 and only one type of particle remains in Λ µ ( F ) .Furthermore, since the collisions are of the form p + p → ∅ , it is clear that for all x ∈ A Z , D p ( F ( x )) − D p ( x ) = D p ( F ( x )) − D p ( x ) . Therefore, which particle remains is decidedaccording to whether µ ([00]) > µ ([11]) or the opposite, both particles disappearing in case ofequality. The third case follows from the fact that if , / ∈ L (Λ µ ( F )) , then Λ µ ( F ) = { ∞ ∞ , ∞ ∞ } which support a unique measure (cid:99) δ . (cid:3) n -state cyclic automaton The n -state cyclic automaton C n is a cellular automaton defined on the alphabet A = Z /n Z by thelocal rule c n ( x i − , x i , x i +1 ) = (cid:26) x i + 1 if x i − = x i + 1 or x i +1 = x i + 1; x i otherwise.See Figure 1 for an example of space-time diagram. his automaton was introduced by [Fis90b]. In this paper, the author shows that for all Bernoullimeasure µ , the set [ i ] (for i ∈ A ) is a µ -attractor iff n ≥ : that is, µ ( { x ∈ A Z : ∃ T ∈ N , ∀ t ≥ T, F t x ∈ [ i ] } ) > for all i . Simulations starting from a random configuration suggest the following:for n = 3 or , monochromatic regions keep increasing in size; for n ≥ , we observe the convergenceto a fixed point where small regions are delimited by vertical lines. We use the main result toexplain this observation. Proposition 3.
Define: u + = { ab ∈ A : ( b − a ) mod n = +1 } ; u − = { ab ∈ A : ( b − a ) mod n = − } ; u = { ab ∈ A : ( b − a ) mod n / ∈ {− , , }} . Then, for any measure µ ∈ M σ − erg (( Z /n Z ) Z ) , only one of those three sets may intersect the languageof Λ µ ( C n ) .If furthermore µ is a Bernoulli measure, then the persisting set can only be u .Proof. We consider the interface defects relatively to the decomposition
Σ = (cid:70) i ∈A Σ i , where Σ i = { ∞ i ∞ } . Σ is a C n -invariant SFT of radius r = 2 , and defects are exactly transitions between colours.Thus we define P = { p ab : ab ∈ A , a (cid:54) = b } . One cell is enough to distinguish the domains ( α = 1 )and we obtain a morphism π of radius defined by the local rule: A → P ∪ { } a · a (cid:55)→ a · b (cid:55)→ p ab for all a, b ∈ A . Simulations suggest that p ab evolves at constant speed +1 if ab ∈ u + , − if ab ∈ u − and if ab ∈ u . Particles progress at their assigned speed unless they meet another particle, in which casethey interact according to the following chemistry: • p ab + p ba → ∅ (if p ab has speed +1); • p ab + p bc → p ac (if p ab and p bc have speeds (+1 , or (0 , − , only when n ≥ ), or • p ab + p bc + p cd → p ad . (if p ab , p bc and p cd have speeds +1 , , − respectively, which is onlypossible for n = 4 ).We group together the particles of same speed, writing p + = { p ab : ab ∈ u + } and p − and p similarly. Formally, for x ∈ A Z and k ∈ Z the update function is defined as: φ ( x, k ) = { k + 1 } if π ( x ) k ∈ p + and (cid:26) π ( x ) k +1 ∈ p + , or π ( x ) k +1 / ∈ P and π ( x ) k +2 / ∈ p − ; { k − } if π ( x ) k ∈ p − and (cid:26) π ( x ) k − ∈ p − , or π ( x ) k − / ∈ P and π ( x ) k − / ∈ p + ; { k } if π ( x ) k ∈ p and π ( x ) k +1 / ∈ p − and π ( x ) k − / ∈ p + ∅ otherwise (and in particular if π ( x ) k = 0) . As previously, we can check that the update function actually describes the dynamics of the particles.For all x ∈ Z , we check that: π ( F ( x )) ∈ p + ⇔ F ( x ) { , } = ab with a = b + 1 ⇔ x [0 , ∈ abbc where c (cid:54) = aabc _ where b = c + 1 dacb where d (cid:54) = a + 1 and c = b − ⇔ (cid:26) π ( x ) ∈ p + and π ( x ) / ∈ P and π ( x ) / ∈ p − , or π ( x ) ∈ p and π ( x ) ∈ p − with good chemistry ( p a,a − + p a − ,a − → p a,a − ) nd so on for other particle types, from which we deduce the hypotheses of Corollary 1. Since [ p + ] = π ([ u + ]) and so on, we obtain the result. If µ is a Bernoulli measure: Consider the “mirror” map γ (( a k ) k ∈ Z ) = ( a − k ) k ∈ Z . γ is continu-ous, and thus measurable. We have µ ( γ ([ u ])) = µ ([ u − ]) = µ ([ u ]) , where ( u · · · u n ) − = u n · · · u .But π ( x ) k ∈ p + ⇔ π ( γ ( x )) − k ∈ p − , and conversely; since F ◦ γ = γ ◦ F , all measures F t ∗ µ are γ -invariant, and thus no particle in p + or p − can persist in L ( π (Λ µ ( F ))) (since otherwise, thesymmetrical particle would persist too). (cid:3) For small values of n or particular initial measures, this proposition can be refined in the followingmanner: n = 3 : p is empty. Given the combinatorics of collisions, where a particle in p + can onlydisappear by colliding with a particle in p − , we see that particles in p + persist if and onlyif π ∗ µ ([ p + ]) > π ∗ µ ([ p − ]) , and symmetrically. In the equality case (in particular, for anyBernoulli measure), no defect can persist in the µ -limit set, which means that Λ µ ( F ) is aset of monochromatic configurations. n = 4 : If µ is a Bernoulli measure, the result of [Fis90b] shows that [ i ] cannot be a µ -attractorfor any i . In other words, for µ -almost all x , F t ( x ) does not converge, which means thatparticles in p + or p − cross the central column infinitely often (even though their probabilityto appear tends to ). This could not happen if particles in p were persisting in π (Λ µ ( F )) ,and thus Λ µ ( F ) is a set of monochromatic configurations. n ≥ : If µ is a nondegenerate Bernoulli measure, the result of [Fis90b] shows that [ i ] is a µ -attractor for all i . This means that some particles in p persist in π (Λ µ ( F )) , and anyconfiguration of Λ µ ( F ) contains only homogeneous regions separated by vertical lines.For n = 3 or , since Λ µ ( F ) is a set of monochromatic configurations we deduce that the sequence ( F n µ ) n ∈ N converges to a convex combination of Dirac measures. However this method does notgive any insight as to the coefficient of each component. As shown in [HdM14], if µ is a Bernoullimeasure then C t ∗ µ −→ t →∞ µ ([2]) (cid:98) δ + µ ([0]) (cid:98) δ + µ ([1]) (cid:98) δ . The problem is open for the 4-cyclic cellular automaton.2.6.3.
One-sided captive cellular automata
We consider the family of captive cellular automata F : A Z → A Z of neighbourhood { , } , whichmeans that the local rule f : A { , } → A satisfies f ( a a ) ∈ { a , a } . See Figure 1 for an exampleof space-time diagram. Proposition 4.
Let F be a one-sided captive automaton and µ ∈ M σ − erg ( A Z ) . Define: u + = { ab ∈ A : a (cid:54) = b, f ( a, b ) = a } u − = { ab ∈ A : a (cid:54) = b, f ( a, b ) = b } Then either u + ∩ L (Λ µ ( F )) = ∅ or u − ∩ L (Λ µ ( F )) = ∅ .If moreover, for all a, b ∈ A , the local rule satisfies f ( ab ) = f ( ba ) and µ is a Bernoulli measure,then Λ µ ( F ) ⊆ { ∞ a ∞ : a ∈ A} (no particle remains).Proof. We consider the interface defects relative to the decomposition
Σ = (cid:70) i ∈A Σ i where Σ i = { ∞ i ∞ } and obtain the same particles P and morphism π as the n -state cyclic automata. p ab evolve t speed − if f ( a, b ) = b and if f ( a, b ) = a , and we define p − and p accordingly. The updatefunction is defined as follows: ∀ x ∈ A Z , ∀ k ∈ Z , φ ( x, k ) = { k } if π ( x ) k ∈ p and π ( x ) k +1 / ∈ p − { k − } if π ( x ) k ∈ p − and π ( x ) k − / ∈ p ∅ otherwiseAs in the two previous examples, we check by enumeration of cases that the update function describesthe particle dynamics on all words of length 3: ∀ x ∈ A Z , ∀ k ∈ Z , F ( x ) k ∈ p ⇔ F ( x ) [ k,k +1] = ab where a (cid:54) = b and f ( a, b ) = a ⇔ x [ k,k +1] = abc where b = c or f ( b, c ) = b ⇔ π ( x ) k ∈ p and π ( x ) k +1 / ∈ p − and deduce the properties of locality, redistribution, disjunction, coalescence and speed fromthere. The main result implies the theorem. If µ is a Bernoulli measure: Then µ is invariant under the mirror map γ and F ◦ γ = γ ◦ F by hypothesis. As in the previous example, we conclude that no particle can persist in Λ µ ( F ) . (cid:3) An automaton performing random walks
Let F be defined on the alphabet A = ( Z / Z ) on the neighbourhood {− , . . . , } by the local rule f defined as follows: f : ( a − , b − ) , . . . , ( a , b ) (cid:55)→ ( a − + a , c ) where c = 1 if ( a − , b − ) = (1 , or ( a , b ) = (0 , otherwise.Intuitively, the first layer performs addition mod at distance 2, while the ones on the second layerbehave as particles, moving right if the first layer contains a 1 and not moving if it contains a 0.Two colliding particles simply merge. Figure 6.
Automaton performing random walks iterated on the uniform measure. (cid:4) is a particle, while the second layer is represented by (cid:3) (0) or (cid:4) (1).
Proposition 5.
Let ν ∈ M σ − erg (( Z / Z ) Z ) and µ = λ × ν , where λ is the uniform measure on ( Z / Z ) Z . Then F t ∗ µ −→ t →∞ λ × (cid:98) δ .Proof. Pivato’s formalism is not necessary here. Consider the set of particles P = { } and themorphism π that is the projection on the second layer. The update function is defined as: ∀ x ∈ A Z , ∀ k ∈ Z , φ ( x, k ) = { k + 1 } if x k = (1 , { k } if x k = (0 , ∅ otherwise. hecking locality, redistribution, disjunction and coalescence is trivial here. Intuitively, each particleperforms a random walk with independent steps and bias . Thus Corollary 1 is not sufficient toconclude, and we need to use the general result of Theorem 1 by proving that { } clashes with itself.Writing ( a tk , b tk ) = F t ( x ) k , we have a tk = (cid:80) tn =0 (cid:0) tn (cid:1) a k − t +4 n mod 2 by straightforward induction.Now take some configuration x with a particle at position k and consider φ t ( x, k ) the walk performedby the particle. First we prove that the particle performs a random walk as claimed above. Wehave: φ t +1 ( x, k ) − φ t ( x, k ) = a tφ t ( x,k ) = (cid:32) t (cid:88) n =0 (cid:18) nt (cid:19) a φ t ( x,k ) − t +4 n (cid:33) mod 2= (cid:32) a φ t ( x,k ) − t + t − (cid:88) n =1 (cid:18) nt (cid:19) a φ t ( x,k ) − t +4 n + a φ t ( x,k )+2 t (cid:33) mod 2 . In the last line, we isolated the leftmost and rightmost term. Since φ t ( x, k ) − t is strictly decreasingand φ t ( x, k ) + 2 t is strictly increasing in t , these terms do not appear in any φ t (cid:48) +1 ( x, k ) − φ t (cid:48) ( x, k ) for t (cid:48) < t . Therefore, if x is drawn according to a Bernoulli measure, the value of a φ t ( x,k ) ± t isindependent of the value of all φ t (cid:48) +1 ( x, k ) − φ t (cid:48) ( x, k ) for t (cid:48) < t .Formally, the behaviour of φ t (cid:48) ( x, k ) (for t (cid:48) ≤ t ) only depends on the random variables { a n : φ t ( x, k ) − t + 1 ≤ n ≤ φ t ( x, k ) + 2 t − } . Let U be any event in the sigma-algebra generated bythese variables. Then we have: µ (cid:0) φ t +1 ( x, k ) − φ t ( x, k ) = 0 | U (cid:1) = µ (cid:32) a φ t ( x,k ) − t = 0 ∧ t (cid:88) n =1 (cid:18) nt (cid:19) a φ t ( x,k ) − t +4 n = 0 | U (cid:33) + µ (cid:32) a φ t ( x,k ) − t = 1 ∧ t (cid:88) n =1 (cid:18) nt (cid:19) a φ t ( x,k ) − t +4 n = 1 | U (cid:33) = µ (cid:16) a φ t ( x,k ) − t = 0 (cid:17) · µ (cid:32) t (cid:88) n =1 (cid:18) nt (cid:19) a φ t ( x,k ) − t +4 n = 0 | U (cid:33) + µ (cid:16) a φ t ( x,k ) − t = 1 (cid:17) · µ (cid:32) t (cid:88) n =1 (cid:18) nt (cid:19) a φ t ( x,k ) − t +4 n = 1 | U (cid:33) = 12 , where the second step is by independence of the leftmost term from all the other variables, andthe third step uses µ (cid:16) a φ t ( x,k ) − t = 0 (cid:17) = since µ is the uniform Bernoulli measure on the firstcomponent. We proved that ( φ t ( x, k )) t ∈ N is a random walk with independent steps and bias .To apply the theorem, we now prove that the particle clashes with itself. Note that the randomwalks performed by different particles are not independent; however, we prove that they are pairwiseindependent.Let k ∈ N . We prove that, when x is chosen according to µ k the conditional measure of µ relativeto the event π ( x ) = π ( x ) k = 1 , φ t ( x, k ) − φ t ( x, performs an unbiased and independent random alk with a “death condition” on 0 (particle collision). Consider the evolution of φ t ( x, k ) − φ t ( x, at each step: δ t ( x ) = ( φ t +1 ( x, k ) − φ t ( x, k )) − ( φ t +1 ( x, − φ t ( x, (cid:32) t (cid:88) n =0 (cid:18) nt (cid:19) a φ t ( x,k ) − t +4 n mod 2 (cid:33)(cid:124) (cid:123)(cid:122) (cid:125) T − (cid:32) t (cid:88) n =0 (cid:18) nt (cid:19) a φ t ( x, − t +4 n mod 2 (cid:33)(cid:124) (cid:123)(cid:122) (cid:125) T Note that the leftmost term of T1 is independent from T2 and all the past values of φ t +1 ( x, k ) − φ t ( x, k ) ; similarly, the rightmost term of T2 is independent from T1 and all past values of φ t +1 ( x, − φ t ( x, . By the same argument as above, T1 and T2 are each worth or with probability independently of each other and of all values of δ t (cid:48) for t (cid:48) < t . We conclude that δ t takes values − , , +1 with probability , , respectively independently of all values of δ t (cid:48) for t (cid:48) < t .Therefore φ t ( x, k ) − φ t ( x, performs an unbiased and independent random walk. This impliesthat µ k ( { x : ∀ t, φ t ( x, k ) > φ t ( x, } ) = 0 (standard result in one-dimensional random walks). Sinceparticles cannot cross, they almost surely end up being in interaction, and therefore { } clashes withitself µ -almost surely. Applying the theorem, we find that no particle can remain in Λ µ ( F ) .More precisely, if we write π i the morphism projecting on the i -th coordinate, π ∗ F t ∗ µ → (cid:98) δ . Sincethe addition mod 2 automaton is surjective, it leaves the uniform measure invariant. Therefore π ∗ F t ∗ µ = λ , and we conclude that F t ∗ µ → λ × (cid:98) δ . (cid:3) Probabilistic cellular automata
Adaptation of our formalism for probabilistic cellular automata
This approach can be adapted to non-deterministic cellular automata, and in particular probabilisticcellular automata. We use here a generalised version of the standard definition.
Definition 5.
Let A be a finite alphabet and N ⊂ Z . We define a map that applies a bi-infinitesequence of local rules to a configuration componentwise: Φ N : ( A A N ) Z × A Z → A Z (( f i ) i ∈ Z , ( x i ) i ∈ Z ) (cid:55)→ ( f i (( x i + r ) r ∈N ) i ∈ Z . Definition 6 (Generalised probabilistic cellular automata) . A generalised probabilistic cellular au-tomaton ˜ F on the alphabet A with neighbourhood N is defined by a measure on bi-infinite sequenceof local rules ν ∈ M σ (( A A N ) Z ) .For a configuration x ∈ A Z , ˜ F : A Z → M σ ( A Z ) is then defined as:For any Borel set U, ˜ F ( x )( U ) = (cid:90) ( A AN ) Z U (Φ N ( f, x ))d ν ( f ) . A deterministic cellular automaton F defined by a local rule f corresponds in this formalismto a Dirac ν = (cid:98) δ f (in which case the image measure is a Dirac on the image configuration), andusual probabilistic cellular automata correspond to the case where ν is a Bernoulli measure; in otherwords, the local rule that applies at each coordinate is drawn independently among a finite set oflocal rules A N → A . Definition 7 (Action on the space of measures) . A generalised probabilistic cellular automatondefined by a measure ν ∈ M σ (( A A N ) Z ) extends naturally to an action ˜ F ∗ : M σ ( A Z ) → M σ ( A Z ) by defining ˜ F ∗ µ ( U ) = (cid:90) A Z (cid:90) ( A AN ) Z U (Φ N ( f, x ))d ν ( f )d µ ( x ) . he µ -limit measures set of ˜ F , V ( ˜ F , µ ) , is the set of cluster points of the sequence ( ˜ F ∗ t µ ) t ∈ N , andthe µ -limit set can be defined as Λ µ ( ˜ F ) = (cid:91) η ∈V ( ˜ F ,µ ) supp η. The definitions of a particle system extend directly, except that the update function also dependson the choice of the local rules as well as on the configuration. Therefore we write φ ( x, n, ( f i )) instead of φ ( x, n ) , where x ∈ A Z , n ∈ Z and ( f i ) ∈ ( A A N ) Z , and the composition notation issimplified as follows (inductively): φ t (cid:16) x, n, ( f k ) ≤ k For any real p ∈ [0 , , consider the probabilistic automaton ˜ F on the alphabet { , } defined onthe neighbourhood N = {− , , } by local rules drawn independently between the traffic rule (rule p and the majority rule (rule F ( x ) i = 1 if and only if x − + x + x ≥ ) with probability − p . This corresponds to the case where ν is aBernoulli measure. igure 7. Example of probabilistic cellular automata where the update of each cellis chosen between two one sided captive CA. p = p = p = Figure 8. Dynamics of the traffic-majority automaton iterated on the initial mea-sure Ber( , ) . Density classification is more efficient with p close to 1.This automaton was introduced by Fatès in [Fat13] as a candidate to solve the density classifica-tion problem.In [BFMM13], the authors completely describe the invariant measures of this PCA. In the conti-nuity of the rest of the article, we are interested in the convergence properties of all σ -ergodic initialmeasure. None of these two results imply the other. Proposition 6 (Prop. 5.5 in [BFMM13]) . For any p in [0 , , the set of ˜ F -invariant measures isthe set of convex combinations of (cid:98) δ , (cid:98) δ and (cid:99) δ . Proposition 7. Let µ ∈ M σ − erg ( A Z ) and p be a real in [0 , . Then Λ µ ( ˜ F ) ⊂ { ∞ ∞ , ∞ ∞ , ∞ (01) ∞ , ∞ (10) ∞ } . As a consequence, any µ -limit measure of ( ˜ F t ∗ µ ) t ∈ N is a convex combination of (cid:98) δ , (cid:98) δ and (cid:99) δ .Proof. The cases p = 0 , correspond to deterministic automata and can be treated easily. he visual intuition suggests to consider interface defects according to the decomposition Σ (cid:116) Σ (cid:116) Σ , where Σ = { ∞ ∞ } , Σ = { ∞ ∞ } (monochromatic subshifts) and Σ = { ∞ (01) ∞ , ∞ (10) ∞ } (chequerboard subshift), since those SFTs are invariant under the action of both rules. The set ofparticles would be P = { p i,j : i (cid:54) = j ∈ { , , }} .However, as Figure 9 shows, the particle p can “explode” and give birth to two particles p and p , contradicting the condition of coalescence. To solve this problem, we tweak the particle systemby replacing each particle p by one particle p and one particle p . “explosion” p p p + p = ∅ p + p = p p Figure 9. Fatès’ traffic-majority probabilistic automaton, with p = .The corresponding morphism π is defined on the neighbourhood { , . . . , } by the local rule: (cid:55)→ p _ (cid:55)→ p (cid:55)→ p _ (cid:55)→ p (cid:55)→ p otherwise (cid:55)→ where the wildcards _ can take both values.Empirically, the particle behaviour without interactions is as follows. Regardless of the rule thatis applied, p , p and p move at a constant speed , +1 and − respectively. A particle p moves at speed − if rule is applied at its position and at speed +1 otherwise (independentrandom walk with bias − p ), except if a particle p prevents its movement to the right, in whichcase it does not move. The particle p behaves symmetrically. As an abuse of notation, we denotefor easier reading π ( x ) = p − if π ( x ) = p and f k = and so on.Particle interactions are of the form p ij + p ji → ∅ , p ij + p jk + p ki → ∅ , or p ij + p jk → p ik , althoughsome of these can not happen. Interactions involve particles at distance at most 3. ormally, we prove through exhaustive case enumeration of all patterns of length 7 and possiblelocal rules that: π ( F ( x )) = p ⇐⇒ ( π ( x ) = p and π ( x ) − (cid:54) = p +20 and π ( x ) (cid:54) = p − ) p moves at speed or ( π ( x ) − = p and π ( x ) = p ) p + p → p π ( F ( x )) = p ⇐⇒ ( π ( x ) − = p and π ( x ) / ∈ { p , p − } ) p moves at speed +1 or ( π ( x ) = p and π ( x ) = p − and π ( x ) − (cid:54) = p +20 ) p + p → p π ( F ( x )) = p ⇐⇒ ( π ( x ) − = p +12 and π ( x ) / ∈ { p , p − } ) p +12 moves at speed +1 or ( π ( x ) = p +12 and π ( x ) = p − ) p +12 is blockedor ( π ( x ) = p − and π ( x ) − (cid:54) = p ) p − moves at speed − π ( F ( x )) = p ⇐⇒ ( π ( x ) − = p +20 and π ( x ) (cid:54) = p ) p +20 moves at speed +1 or ( π ( x ) = p − and π ( x ) − = p +12 ) p − is blockedor ( π ( x ) = p − and π ( x ) − (cid:54) = p and π ( x ) = p +12 ) p − moves at speed − π ( F ( x )) = p ⇐⇒ ( π ( x ) = p and π ( x ) − (cid:54) = p and π ( x ) (cid:54) = p +12 ) p moves at speed − or ( π ( x ) − = p +20 and π ( x ) = p and π ( x ) (cid:54) = p − ) p + p → p π ( F ( x )) = 0 in all other cases (including other possible interactions)Using this statement, it is straightforward though tedious to define formally the update function,and the various conditions of locality, disjunction, particle control, surjectivity and coalescence areproved similarly to the previous examples.Assume p ≥ . We show that no particle can remain asymptotically by applying the main resulton the sets ( P i ) ≤ i ≤ : { p } , { p } , { p } , { p } and { p } . We need only to show the clashesrelative to the second and fourth sets since all other clashes are consequences of the speed of theseparticles.Let k ∈ N and x be such that π ( x ) = p and π ( x ) k ∈ { p , p } . Since p progresses at speed1, the distance φ t ( x, k ) − φ t ( x, cannot increase, and it decreases by at least one with probability p (respectively − p ). It is clear that the particles end up in interaction ν ∞ -almost surely. Showingthat p and p clash with p is symmetric.Let x be such that π ( x ) = p and π ( x ) k = p . As long as there are no interactions, thedistance φ t ( x, k ) − φ t ( x, 0) = − φ t ( x, performs an independent random walk of bias p − , wherea increasing step is sometimes replaced by a constant step. Such a random walk reaches ν ∞ -almost surely, which shows that the particles end up in interaction. The clashes between p and p , and between { p } and { p } , are proved in a similar manner. The same proof holds for p ≤ by exchanging the roles of p and p .Applying Theorem 2, we conclude that only one particle p ij can remain in the µ -limit set. Thisresult can be improved further: consider V k = { x ∈ Λ µ ( F ) : π ( x ) k = p ij } . Configurations in V k areof the form y · z , where y ∈ A ] −∞ ,k ] is admissible for Σ i and z ∈ A [ k +1 , + ∞ [ is admissible for Σ j ;in particular, they contain only one particle. For any measure η ∈ V ( ˜ F , µ ) , η ( V k ) is independentfrom k by σ -invariance, and η ( (cid:83) k V k ) = (cid:80) k η ( V k ) ≤ by disjunction of the ( V k ) k ∈ Z . Consequently, η ( V k ) = 0 , which means V k / ∈ supp( η ) . We conclude that no particle remain in the µ -limit set, orin other words, Λ µ ( F ) ⊂ Σ ∪ Σ ∪ Σ . (cid:3) .7.3. Example: Approximation of a line A finite word of { , } ∗ can be seen as a finite curve in Z taking its origin in (0 , , moving right ona and up on a . In [RR15], the authors introduce for any α ∈ Q ∩ [0 , a random process that,starting from a finite word w ∈ { , } ∗ whose frequency of symbols is α , organises bits throughlocal flips to obtain asymptotically a discrete segment of slope α .We adapt these processes so that the flips are performed in parallel and on an infinite configu-ration, which gives a probabilistic cellular automaton for every slope α ∈ Q ∩ [0 , . We considerthe action of this PCA on any initial σ -ergodic measure satisfying µ ([1]) = α . Using Theorem 2,we show that the sequence of measures converges towards the measure supported by a periodicconfiguration representing a discrete line of slope α . To simplify the presentation, we consider herethat α = ; the method can be easily generalised to other slopes.Define the following local rules: • i is the identity; • r ( x − , x − , x , x ) = (cid:40) x if x − x − x x = 0101 or ,x − otherwise ; • (cid:96) ( x − , x , x , x ) = (cid:40) x if x − x x x = 0101 or ,x otherwise . Let ˜ F line be a probabilistic cellular automaton (represented in Figure 10) defined by a σ -ergodicmeasure ν ∈ M σ ( { g , g , g − } Z ) whose support is the subshift of finite type defined by the set offorbidden patterns { ir, (cid:96)(cid:96), (cid:96)i, rr, r(cid:96) } . To put it more simply, any time the local rules in two consecutive cells are (cid:96) and r (which happenswith positive probability), the probabilistic CA permutes these two letters, except if they are at thecentre of a four-letter words or . In any other situation, it acts as the identity. Proposition 8. Let µ ∈ M σ − erg ( A Z ) . Then: µ ([00]) > µ ([11]) ⇒ / ∈ Λ µ ( ˜ F line ); µ ([00]) < µ ([11]) ⇒ / ∈ Λ µ ( ˜ F line ); µ ([00]) = µ ([11]) ⇒ ˜ F t line µ → (cid:99) δ . Proof. We consider the dislocation defects with regards to the chequerboard SFT Σ = { ∞ (01) ∞ , ∞ (10) ∞ } .As in Section 2.6.1, we obtain the particles (cid:55)→ p and (cid:55)→ p . A particle p at position moves at speed +2 if (cid:96) is applied at position , at speed − if (cid:96) is applied on position − , andat speed otherwise. The particle p is symmetrical and they annihilate on contact. Indeed, wecheck by straightforward case enumeration that: π ( ˜ F line ( x )) = p ⇔ ˜ F line ( x ) [0 , = 00 ⇔ x [0 , = 00 and f − , x − (cid:54) = ( (cid:96), , ( f , x ) (cid:54) = ( (cid:96), x [ − , − = 00 and f − = (cid:96), x = 0 x [2 , = 00 and f = (cid:96), x = 0 ⇔ π ( x ) = p at speed π ( x ) − = p at speed + 2 and π ( x ) (cid:54) = p π ( x ) = p at speed − and π ( x ) (cid:54) = p , nd similarly for p . From this we deduce the various hypotheses of theorem, including ν ∞ -almostsure clashing which stems from the fact that particles perform random walks. The exact statementof the proposition follows through the same arguments as in Section 2.6.1; in particular, the thirdcase corresponds to the discrete line of slope . (cid:3) Figure 10. Example of a space-time diagram of ˜ F line , where ν is the Markov mea-sure maximising the entropy of the subshift of finite type defined by the forbiddenpatterns { f f − , f f , f f , f − f − , f − f } . Particle-based organisation: quantitative results For some cellular automata with simple defect dynamics, the previous results can be refined with aquantitative approach: that is, to determine the asymptotic distribution of random variables relatedto the particles. In [KFD11], P. Kůrka, E. Formenti and A. Dennunzio considered T n ( x ) , the entrytime after time n on the initial configuration x , which is the waiting time before a particle appearsin a given position after time n . They restricted their study to a gliders automaton, which is acellular automaton on 3 states: a background state and two particles evolving at speeds 0 and -1that annihilate on contact. Thus, we have one entry time for each type of particle ( T + n ( x ) and T − n ( x ) ). When the initial configuration is drawn according to the Bernoulli measure of parameters ( , , ) , which means that each cell contains, independently, a particle of each type with probability , they proved that: ∀ α ∈ R + , µ (cid:18) T − n ( x ) n ≤ α (cid:19) −→ n →∞ π arctan √ α. They also suggested to develop formal tools in order to be able to handle more complex automata,starting with the ( − , symmetric case.In Section 3.2, we extend this result to allow arbitrary values for the particle speeds v − and v + ,and relax the conditions on the initial measure to some mixing conditions. Then, when v − < and v + ≥ , we have: ∀ α ∈ R + , µ (cid:18) T − n ( x ) n ≤ α (cid:19) −→ n →∞ π arctan (cid:18)(cid:114) − v − αv + − v − + v + α (cid:19) , and symmetrically if we exchange + and − . The proof relies on the fact that the behaviour ofgliders automata can be characterised by some random walk process; this idea was introduced byV. Belitsky and P. Ferrari in [BF95] and was already used in [KM00] and [KFD11]. In our case, aparticle appearing in a position corresponds to a minimum between two concurrent random walks. he new tool here is that under α -mixing conditions, we rescale this process and approximate itwith a Brownian motion. Thus we obtain the explicit asymptotic distribution of entry times.This method, consisting in associating a random walk to each gliders automata and studyingthis random walk using scale invariance, is not limited to this particular conjecture concerningentry times. Indeed, we see in the next two sections that it can be used to study the asymptoticbehaviour of two other, arguably more natural, parameters: the particle density at time t and therate of convergence to the limit measure. However, we obtain only an upper bound instead of anexplicit asymptotic distribution. There is no doubt this method can be adapted to other parametersin a similar way.Furthermore, these results can be extended to other automata with similar behaviour, suchas those in Figure 1, by factorising them onto a gliders automaton. This point is discussed inSection 3.6. This method is more difficult to generalise when there is birth of particle, even in asimple case such as the -cyclic cellular automaton. Gliders automata and random walks In this section we give the definition and the first properties of the class of gliders automata. Definition 8 (Gliders automata) . Let v − < v + ∈ Z . The ( v − , v + ) -gliders automaton (or GA) G isthe cellular automaton of neighbourhood [ − v + , − v − ] defined on the alphabet A = {− , , +1 } bythe local rule: f ( x − v + . . . x − v − ) = +1 if x − v + = +1 and ∀ N ≤ − v − , (cid:80) Nn = − v + +1 x n ≥ − if x − v − = − and ∀ N ≥ − v + , (cid:80) − v − − n = N x n ≤ otherwise.In all the following, A = {− , , +1 } and the diagrams are represented with the convention (cid:3) = 0 , (cid:4) = +1 , (cid:4) = − . Figure 11. Space-time diagram of the ( − , -gliders automaton on a random ini-tial configuration.Our results apply on automata with simple defects dynamics, namely, automata admitting aparticle system with P = {± } and whose update function corresponds to a gliders automaton. Wefirst prove our results for gliders automata before generalising them in Section 3.6. Let us introducesome tools that turn the study of the dynamics of a gliders automaton into the study of somerandom walk. efinition 9 (Random walk associated with a configuration) . Let x ∈ {− , , } Z . Define thepartial sums S x by: S x (0) = 0 and ∀ k ∈ Z , S x ( k + 1) − S x ( k ) = x k . We extend S x to R by piecewise linear interpolation: S x ( t ) = ( (cid:100) t (cid:101) − t ) S x ( (cid:98) t (cid:99) ) + ( t − (cid:98) t (cid:99) ) S x ( (cid:100) t (cid:101) ) for t ∈ R \ Z . We also introduce the rescaled process S kx : t (cid:55)→ S x ( kt ) √ k .This random walk is simpler to study than the space-time diagram of the gliders automaton, andactually contains the same amount of information, as shown by the following technical lemmas. Definition 10. Let f : R → R and U ⊂ R . We define argmin U f by: ∀ t ∈ U, t = argmin U f ⇐⇒ ∀ t (cid:48) ∈ U \{ t } , f ( t ) < f ( t (cid:48) ) . In other words, t realises the strict minimum of f on U ; this point is not always defined. Lemma 2. Let G be the ( v − , v + ) -gliders automaton. For all j ∈ Z and n ≥ , j = argmin [ j, j + n ] S G ( x ) ⇐⇒ j − v + = argmin [ j − v + , j + n − v − ] S x ,j = argmin [ j − n, j ] S G ( x ) ⇐⇒ j − v − = argmin [ j − n − v + , j − v − ] S x . Proof. We prove those equivalences by induction on n . At each step, we prove only the first equiv-alence, the other one being symmetric. Base case: S G ( x ) ( j ) < S G ( x ) ( j + 1) ⇔ G ( x ) j = +1 ⇔ x j − v + = +1 and ∀ N ≤ − v − , N (cid:88) t = − v + +1 x j + t ≥ ⇔ S x ( j − v + ) < min [ j +1 − v + , j +1 − v − ] S x . Induction: Assume both equivalences hold for some n ≥ .Suppose j = argmin [ j, j + n +1] S G ( x ) . In particular j = argmin [ j, j + n ] S G ( x ) , and by induction hypothesis j − v + = argmin [ j − v + , j + n − v − ] S x . We distinguish two cases: • if S x ( j + n − v − + 1) > S x ( j − v + ) , then j − v + = argmin [ j − v + , j + n − v − +1] S x and we conclude; • otherwise, this means that S x ( j + n − v − + 1) = S x ( j − v + ) (the walk can decrease byat most one at each step), and thus j + n − v − + 1 = argmin [ j − v + +1 , j + n − v − +1] S x . By induction hypothesis, j + n + 1 = argmin [ j +1 , j + n +1] S G ( x ) , and in particular S G ( x ) ( j + n + 1) < S G ( x ) ( j + 1) . Therefore S G ( x ) ( j + n + 1) ≤ S G ( x ) ( j ) ,a contradiction with the first assumption.The converse is proved in a similar manner. (cid:3) emma 3. Let G be the ( v − , v + ) -gliders automaton. For all j ∈ Z and k ≥ ,G t ( x ) j = − ⇐⇒ j − v − t + 1 = argmin [ j − v + t, j − v − t +1] S x G t ( x ) j = +1 ⇐⇒ j − v + t = argmin [ j − v + t, j − v − t +1] S x . This is illustrated in Figure 12. S x j − k + 1 j + k a k jG k ( x ) j Figure 12. Illustration of Lemma 3. A strict minimum is reached on j − k + 1 . Proof. By induction on t , proving only the first equivalence at each step: Base case ( t = 0) : By definition of S x , S x ( j + 1) < S x ( j ) ⇔ x j = − . Induction: Assume that both equivalences hold for a given time t . By applying the inductionhypothesis on G ( x ) , G t +1 ( x ) j = − ⇔ j − v − t + 1 = argmin [ j − v + t, j − v − t +1] S G ( x ) and we concludeby applying Lemma 2. (cid:3) Entry times The main result of Section 2 implies that, for any σ -ergodic initial measure µ , Λ G ( µ ) contains atmost one kind of particle, which one depending on whether µ ([+1]) > µ ([ − or the opposite.When µ ([+1]) = µ ([ − , Λ G ( µ ) only contains the particle-free configuration ∞ ∞ . In other words, G t ∗ µ → (cid:98) δ , which means that the probability of seeing a particle in any fixed finite window tends to0 as t → ∞ . Definition 11 (Entry times) . Let v − < ≤ v + ∈ Z , G the ( v − , v + ) -GA and x ∈ {− , , } Z . Wedefine: T − n ( x ) = min { k ∈ N : ∃ i ∈ [0 , | v − | − , G k + n ( x ) i = − } , with T − n ( x ) = ∞ if this set is empty. This is the entry time in the set { b ∈ {− , , } Z : ∃ i ∈ [0 , | v − | − , b i = − } after time n at position 0 starting from x . We define T + n ( x ) in a symmetricalmanner. − n ( x ) nx Figure 13. An entry time for the (-3,1)-gliders automaton.The size of the considered window is such that any particle “passing through” the column 0appears in this window exactly once (See Figure 13). Of course entry times for particles of speed 0make no sense. From now on, we only consider T − for simplicity, all the results being valid for T + .As a consequence of Birkhoff’s ergodic theorem, when µ ([ − > µ ([+1]) , − particles persist µ -almost surely and their density converges to a positive number. Therefore: • µ ( T + n ( x ) = ∞ ) −→ n →∞ ; • ∀ α > , µ (cid:16) T − n ( x ) n ≤ α (cid:17) −→ n →∞ ,and symmetrically. This is why we only consider the case µ ([ − µ ([+1]) . Kůrka and al.proved the following result: Theorem 3 ([KFD11]) . For the ( − , -GA (“Asymmetric gliders”) with an initial measure µ =Ber (cid:0) , , (cid:1) : ∀ α > , µ (cid:18) T − n ( x ) n ≤ α (cid:19) −→ n →∞ π arctan √ α. In the same article, they conjectured that this result could be extended to any initial Bernoullimeasure of parameters ( p, − p, p ) for ≤ p ≤ by replacing the right-hand term by π arctan √ pα .We will prove that this conjecture is actually incorrect.To state our result, we introduce two particular subclasses of M σ ( A Z ) . We introduce α -mixingcoefficients of a measure µ ∈ M σ ( A Z ) : α µ ( n ) = sup {| µ ( A ∩ B ) − µ ( A ) µ ( B ) | : A ∈ B ] −∞ , , B ∈ B [ n, + ∞ [ } . where B [ a,b ] is the Borel σ -algebra generated by the random variables ( X i ) a ≤ i ≤ b .Define: • Ber = the set of Bernoulli measures on {− , , +1 } Z and parameters ( p, − p, p ) for some < p ≤ ; • M ix the set of measures µ ∈ M σ ( {− , , +1 } Z ) satisfying: – (cid:82) A Z x d µ ( x ) = 0 , i.e., µ ([ − µ ([+1]) ; – (cid:80) ∞ k =0 (cid:82) A Z x · x k d µ ( x ) converges absolutely to a real σ µ > (asymptotic variance); – ∃ ε > , (cid:80) n ≥ α µ ( n ) − ε < ∞ .In particular, Ber = ⊂ M ix . heorem 4 (Quantitative result for entry time) . For any ( v − , v + ) -GA with v − < and v + ≥ and any initial measure µ ∈ M ix , ∀ α > , µ (cid:18) T − n ( x ) n ≤ α (cid:19) −→ n →∞ π arctan (cid:18)(cid:114) − v − αv + − v − + v + α (cid:19) . Notice that this limit is independent from µ (as long as µ ∈ M ix ), disproving the conjecturewhen µ ∈ Ber = . Brownian motion and proof of the main result The third hypothesis for M ix is chosen so that the large-scale behaviour of the partial sums S x ( t ) can be approximated by a Brownian motion. This invariance principle is the core of our proofs.The first and second conditions ensure that the Brownian motion obtained this way have no biasand nonzero variance, respectively. Definition 12 (Brownian motion) . A Brownian motion (or Wiener process ) B of mean andvariance σ is a continuous time stochastic process taking values in R such that: – B (0) = 0 , – t (cid:55)→ B ( t ) is almost surely continuous, – B ( t ) − B ( t ) follow the normal law of mean 0 and variance ( t − t ) σ ; – For t < t ≤ t (cid:48) < t (cid:48) , increments B ( t ) − B ( t ) and B ( t (cid:48) ) − B ( t (cid:48) ) are independent.See [MP10] for a general introduction to Brownian motion. Proposition 9 (Rescaling property) . Let B be a Brownian motion. Then, for any k > , t (cid:55)→ √ k B ( kt ) is a Brownian motion with same mean and variance. We now state some invariance principles, which consists in approximating rescaled random walksby Brownian motion. We use a strong version, which guarantees an almost sure convergence byconsidering a copy of the process in a richer probability space. Theorem 5 ([ZC96], Corollary 9.3.1) . Let X = ( X i ) i ∈ N be a family of random variables takingvalues in {− , , } . We denote by α X ( n ) its α -mixing coefficients defined as: α X ( n ) = sup {| P ( A ∩ B ) − P ( A ) P ( B ) | : t ∈ N , A ∈ B [0 ,t ] , B ∈ B [ t + n, + ∞ [ } , where again B [ a,b ] is the sigma-algebra generated by ( X a , . . . , X b ) .Assume that: (1) ∀ i, E ( X i ) = 0 ; (2) t E (cid:16)(cid:80) (cid:98) t (cid:99) i,j =1 X i · X j (cid:17) converges absolutely to some positive real σ ; (3) ∃ ε > , (cid:80) ∞ n =1 α X ( n ) + ε .Then we can define two processes X (cid:48) = ( X (cid:48) i ) i ∈ N and B on a richer probability space (Ω , P ) suchthat: (1) X and X (cid:48) have the same distribution; (2) B is a Brownian motion of mean 0, variance σ ; (3) for any ε > , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:98) t (cid:99) (cid:88) i =1 X i − B ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O (cid:16) t + ε (cid:17) P -almost surely . orollary 3. Let µ ∈ M ix . For any fixed constants q < r ∈ R , we can define a process X (cid:48) =( X (cid:48) i ) i ∈ Z and a family of processes ( t (cid:55)→ B n ( t )) n ∈ N on a richer probability space (Ω , P ) such that: (1) X (cid:48) has distribution µ ; (2) every B n is a Brownian motion of mean 0 and variance σ µ > ; (3) for any ε > , denoting by S X (cid:48) the piecewise linear function defined by S X (cid:48) (0) = 0 and S X (cid:48) ( k + 1) − S X (cid:48) ( k ) = X (cid:48) k for all k ∈ Z , ∀ n ∈ N , sup t ∈ [ q,r ] (cid:12)(cid:12)(cid:12)(cid:12) S X (cid:48) ( nt ) √ n − B n ( t ) (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:16) n − + ε (cid:17) P -almost surely . Proof. We apply Theorem 5 on ( X i ) i ∈ N , where ( X i ) i ∈ Z is distributed according to µ . Because µ is σ -invariant, this is a stationary process. The first and third conditions are satisfied by definition of M ix . For the second condition, n (cid:88) ≤ i,j ≤ n | E ( X i · X j ) | = 1 n n (cid:88) i =0 n − i (cid:88) j = − i E ( | X · X j − i | ) by stationarity = n (cid:88) i = − n n + 1 − | i | n E ( | X · X i | ) by reordering the sum ∼ n →∞ n (cid:88) i =0 E ( | X · X i | ) −→ n →∞ σ µ , by stationarity and the equivalence criterion for positive series. We obtain two processes X =( X i ) i ∈ N and B on a richer probability space (Ω , P ) such that X has the same distribution as x , B is a Brownian motion of mean 0, variance σ µ , and: ∀ ε > , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:98) t (cid:99) (cid:88) i =1 X i − B ( t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = O t → + ∞ (cid:16) t + ε (cid:17) P -almost surely . Since the variables X i take value in {− , , } , we have for any t (cid:12)(cid:12)(cid:12)(cid:80) (cid:98) t (cid:99) i =1 X i − S X ( t ) (cid:12)(cid:12)(cid:12) < (a staircaseand piecewise linear function having the same values on N ). Therefore: ∀ ε > , (cid:12)(cid:12) S X ( t ) − B ( t ) (cid:12)(cid:12) = O t → + ∞ (cid:16) t + ε (cid:17) P -almost surely . ∀ ε > , ∀ n ∈ N , √ n (cid:12)(cid:12) S X ( tn ) − B ( tn ) (cid:12)(cid:12) = O n →∞ (cid:16) n − + (cid:15) (cid:17) · O t →∞ (cid:16) | t | + (cid:15) (cid:17) P -almost surely . For any r ∈ R , taking the sup for t ∈ [0 , r ] , we obtain: ∀ ε > , ∀ n ∈ N , sup t ∈ [0 ,r ] (cid:12)(cid:12)(cid:12)(cid:12) S X ( tn ) √ n − B ( tn ) √ n (cid:12)(cid:12)(cid:12)(cid:12) = O n →∞ (cid:16) n − + (cid:15) (cid:17) P -almost surely . By rescaling property B n : t (cid:55)→ B ( tn ) √ n is a Brownian motion of same mean and variance as B .To extend the result to negative values, we apply the theorem again to ( x − i − ) i ∈ N , obtaining aprocess X and a Brownian motion B satisfying the same asymptotic bound on t → −∞ . Joiningboth parts, we can see that the process X (cid:48) = . . . X − , X − , X , X . . . have distribution µ and B n : t (cid:55)→ B n ( t ) if t ≥ , B n ( t ) if t < is a Brownian motion. (cid:3) For a survey of invariance principles under different assumptions, see [MR12].We now prove the main theorem. roof of Theorem 4. For any x ∈ {− , , } Z , Lemma 3 applied on the column 0 gives: T − n ( x ) = min (cid:26) k ∈ N | ∃ j ∈ [0 , − v − [ , S x ( − v − ( n + k ) + j + 1) < min [ − v + ( n + k )+ j, − v − ( n + k )+ j ] S x (cid:27) = min (cid:26) k ∈ N | ∃ j ∈ [0 , − v − [ , S x ( − v − ( n + k ) + j + 1) < min [ − v + ( n + k )+ j, − v − n ] S x (cid:27) Indeed, if ( k, j ) is the smallest pair (in lexicographic order) such that S x ( − v − ( n + k ) + j +1) < min [ − v + ( n + k )+ j, − v − n ] S x , then all pairs ( k (cid:48) , j (cid:48) ) < ( k, j ) verify S x ( − v − ( n + k (cid:48) ) + j (cid:48) + 1) ≥ min [ − v + ( n + k )+ j, − v − n ] S x , and therefore S x ( − v − ( n + k ) + j + 1) < min [ − v + ( n + k )+ j, − v − ( n + k )+ j ] S x .Note that if this last condition is reached on k ∈ N , since S x is piecewise linear, it is attained for t as soon as t > k − and reciprocally. Thus: T − n ( x ) = inf (cid:26) t ≥ | ∃ j ∈ [0 , − v − [ , S x ( − v − ( n + t ) + j + 2) < min [ − v + ( n + t )+ j +1 , − v − n ] S x (cid:27) Replacing j by in this expression adds to the infimum a value comprised between and − v − − − v − (remember v − < ). Since the infimum is necessarily an integer, we compensate by taking theinteger part: T − n ( x ) = (cid:22) inf (cid:26) t ≥ | S x ( − v − ( n + t ) + 2) < min [ − v + ( n + t )+1 , − v − n ] S x (cid:27)(cid:23) = (cid:36) inf (cid:40) t ≥ | S nx (cid:18) − v − (cid:18) tn (cid:19) + 2 n (cid:19) < min [ − v + (1+ tn )+ n , − v − ] S nx (cid:41)(cid:37) = (cid:36) n · inf (cid:40) t ≥ | S nx (cid:18) − v − (1 + t ) + 2 n (cid:19) < min [ − v + (1+ t )+ n , − v − ] S nx (cid:41)(cid:37) where S nx is the rescaled process t (cid:55)→ S x ( nt ) √ n . Since S x is -Lipschitz, S nx is √ n -Lipschitz (i.e. | S nk ( t (cid:48) ) − S nk ( t ) | ≤ √ n | t (cid:48) − t | ). Dividing the previous expression by n , using the fact that t − n ≤ (cid:98) nt (cid:99) n ≤ t for all t, n ∈ R × N : µ (cid:18) min [ − v − , − v − (1+ α )] S nx + 4 √ n < min [ − v + (1+ α ) , − v − ] S nx (cid:19) ≤ µ (cid:18) T − n ( x ) n ≤ α (cid:19) (1a) µ (cid:18) T − n ( x ) n ≤ α (cid:19) ≤ µ (cid:18) min [ − v − , − v − (1+ α )] S nx − √ n < min [ − v + (1+ α ) , − v − ] S nx (cid:19) (1b)We now bound the left-hand term of (1a) from below and the right-hand term of (1b) from above.Using Corollary 3, we build a process X (cid:48) and a family of processes ( B n ) n ∈ N on a richer probabilityspace (Ω , P ) such that X (cid:48) is distributed according to µ and the B n are Brownian motions. ∀ n ∈ N , sup [ − v + (1+ α ) , − v − (1+ α )] (cid:12)(cid:12)(cid:12)(cid:12) S X (cid:48) ( nt ) √ n − B n ( t ) (cid:12)(cid:12)(cid:12)(cid:12) = O (cid:16) n − + ε (cid:17) P -almost surely . By symmetry, B ln ( t ) = B n ( − v − − t ) − B n ( − v − ) and B rn ( t ) = B n ( − v − + t ) − B n ( − v − ) are twoindependent Brownian motions on [0 , v − − v + (1 + α )] and [0 , − v − α ] , respectively. Consequently, or any ε > and n large enough: µ (cid:18) min [ − v − , − v − (1+ α )] S nx + ε < min [ − v + (1+ α ) , − v − ] S nx (cid:19) = P (cid:18) min [ − v − , − v − (1+ α )] S nX (cid:48) + ε < min [ − v + (1+ α ) , − v − ] S nX (cid:48) (cid:19) ≥ P (cid:18) min [ − v − , − v − (1+ α )] B n + 2 ε < min [ − v + (1+ α ) , − v − ] B n (cid:19) ≥ P (cid:18) min [0 , − v − α ] B ln + 2 ε < min [0 , − v − + v + (1+ α )] B rn (cid:19) (2a)and a symmetrical upper bound for (1b): µ (cid:18) min [ − v − , − v − (1+ α )] S nx − ε < min [ − v + (1+ α ) , − v − ] S nx (cid:19) ≤ P (cid:18) min [0 , − v − α ] B ln − ε < min [0 , − v − + v + (1+ α )] B rn (cid:19) (2b)We now evaluate the right-hand expression in (2a).For any Brownian motion B and b > , we have by rescaling P (cid:18) min [0 ,b ] B ≥ m (cid:19) = P (cid:18) min [0 , B ≥ m √ b (cid:19) .Furthermore, since B ln and B rn are independent, so are min [0 , B lx and min [0 , B rx . Denote by µ m the lawof the minimum of a Brownian motion on [0 , , which is defined by the density function: R → R t (cid:55)→ e − t if t ≤ , otherwise. (see [MP10]).This means that for any y, z > : P (cid:18) min [0 ,y ] B ln < min [0 ,z ] B rn (cid:19) = (cid:90) −∞ (cid:90) −∞ {√ y · m ≤√ z · m } dµ m ( m ) dµ m ( m )= ( i ) π (cid:90) −∞ (cid:90) √ z · m √ y −∞ e − m e − m dm dm = ( ii ) π (cid:90) π +arctan ( √ yz ) π (cid:90) + ∞ re − r drdθ = 2 π arctan (cid:18)(cid:114) yz (cid:19) (3)(i) by using the law of the minimum of a Brownian motion, (ii) by passing in polar variables. For ε > , a similar calculation gives: P (cid:18) min [0 ,y ] B ln < min [0 ,z ] B rn (cid:19) − P (cid:18) min [0 ,y ] B ln + 2 ε < min [0 ,z ] B rn (cid:19) ≤ π (cid:90) −∞ (cid:90) √ z · m √ y √ z · m − ε √ y e − m e − m dm dm ≤ ε π √ y (cid:90) −∞ e − ym z e − m dm −→ ε → (4)Using (3) and (4), we see that the right-hand term in (2a) converges to π arctan (cid:16)(cid:113) − v − αv + − v − + v + α (cid:17) as ε → . The left-hand term in (2b) can be bounded from above by the same method. We applythis result to (1a) and (1b) by taking ε = √ n (resp. √ n ), and the theorem follows. (cid:3) .4. Particle density Definition 13 (Particle density in a configuration) . The − particle density in x ∈ {− , , } Z isdefined as d − ( x ) = Freq ( − , x ) . d + ( x ) is defined in a symmetrical manner.In all the following, any result on d − also holds for d + by symmetry. Theorem 6 (Decrease rate of the particle density) . Let G be a ( v − , v + ) -GA with initial measure µ ∈ M ix . Then:For µ -almost all x ∈ {− , , } Z , ∀ ε > , d − ( G t ( x )) = O (cid:16) t − + ε (cid:17) If furthermore µ ∈ Ber = :For µ -almost all x ∈ {− , , } Z , d − ( G t ( x )) ∼ t − Proof. When µ ∈ M ix , it is in particular σ -ergodic, and so are its images G t ∗ µ . By Birkhoff’s ergodictheorem, one has d − ( G t ( x )) = G t ∗ µ ([ − µ ( G t ( x ) = − for µ -almost all x ∈ {− , , } Z .We first prove the theorem when G is the ( − , -gliders automaton. By Lemma 3, µ ( G t ( x ) = − 1) = µ (cid:18) S x ( t + 1) < min [0 ,t ] S x (cid:19) . Equivalence ( µ ∈ Ber = ): By symmetry, µ (cid:18) S x ( t + 1) < min [0 ,t ] S x (cid:19) = µ (cid:18) S x (0) < min [1 ,t +1] S x (cid:19) , which is the probability that the random walk starting from 0 remains strictly positive during t steps, also known as its probability of survival. According to [Red01], when the random walk issymmetric and the steps are independent, we have the equivalence µ ( G t ( x ) = − ∼ √ t . Upper bound: µ (cid:18) S x ( t + 1) < min [0 ,t ] S x (cid:19) ≤ µ (cid:18) S t +1 x (1) = min [0 , S t +1 x (cid:19) . Using Corollary 3, we have: µ (cid:18) S t +1 x (1) = min [0 , S t +1 x (cid:19) = P (cid:18) S t +1 X (cid:48) (1) = min [0 , S t +1 X (cid:48) (cid:19) ≤ P (cid:18) B t +1 (1) ≤ min [0 , B t +1 + C t +1 (cid:19) ≤ P (cid:18) B t +1 (0) ≤ min [0 , B t +1 + C t +1 (cid:19) , where C t +1 = sup [0 , (cid:12)(cid:12) S t +1 X (cid:48) − B t +1 (cid:12)(cid:12) = O (cid:16) t − + ε (cid:17) P -almost surely, and where the third line is obtainedby symmetry of the Brownian motion.Furthermore P (cid:18) min [0 , B t +1 > − C t +1 (cid:19) = (cid:90) − C t +1 e − x / dx ≤ C t +1 = O (cid:16) t − + ε (cid:17) . General case (any v − < v + ): Let G (cid:48) be the ( v − , v + ) -GA. Then G (cid:48) = σ v + ◦ G v + − v − . To conclude, it is enough to see that the particle density is σ -invariant and decreasing under theaction of G . (cid:3) .5. Rate of convergence In this section, we estimate the rate of convergence to the limit measure. For that we fix a distanceon the space M σ ( A Z ) of σ -invariant measures, which induces the weak ∗ topology: d M ( µ, ν ) = (cid:88) n ∈ N n max u ∈A n | µ ([ u ]) − ν ([ u ]) | . Theorem 7 (Rate of convergence to the limit measure) . Let G be the ( v − , v + ) -GA with initialmeasure µ ∈ M ix . Then: ∀ ε > , d M ( G t ∗ µ, (cid:98) δ ) = O (cid:16) t − / ε (cid:17) If furthermore µ ∈ Ber = : d M ( G t ∗ µ, (cid:98) δ ) = Ω (cid:16) t − / (cid:17) Proof. We first prove the theorem when G is the ( − , -gliders automaton. By defining (cid:96) ∈ A (cid:96) the word containing only zeroes, the distance can be rewritten: ∀ t ∈ N , d M ( G t ∗ µ, (cid:98) δ ) = ∞ (cid:88) (cid:96) =1 (cid:96) G t ∗ µ (cid:16) A Z \ [0 (cid:96) ] (cid:17) . Lower bound when µ ∈ Ber = : d M ( G t ∗ µ, (cid:98) δ ) > G t ∗ µ (cid:0) A Z \ [0] (cid:1) . We conclude with Theorem 6. Upper bound: We give an upper bound for G t ∗ µ ( A Z \ [0 (cid:96) ]) = µ ( ∃ ≤ d ≤ (cid:96), G t ( x ) d = ± for (cid:96) ∈ N and t ∈ N . By Lemma 3, ∀ d ∈ Z , G t ( x ) d = +1 ⇔ S x ( d ) < min [ d +1 ,d + t ] S x . Therefore: G t ∗ µ (cid:32) (cid:96) (cid:92) d =0 [+1] d (cid:33) ≤ µ (cid:18) min [0 ,(cid:96) ] S x < min [ (cid:96) +1 ,t ] S x (cid:19) ≤ µ (cid:18) min [0 ,t ] S x ≥ − (cid:96) (cid:19) ≤ µ (cid:18) min [0 , S tx ≥ − (cid:96) √ t (cid:19) By Corollary 3, using the same notations as in the previous proofs: G t ∗ µ ( ∃ ≤ d ≤ (cid:96), x d = +1) ≤ P (cid:18) min [0 , S tX (cid:48) ≥ − (cid:96) √ t (cid:19) ≤ P (cid:18) min [0 , B t ≥ − (cid:96) √ t − C t (cid:19) where C t = O (cid:16) t − + ε (cid:17) = O (cid:16) t − + ε (cid:17) for any ε > , following the same calculations as in Section 3.4. The case of − particles issymmetrical, and we conclude. General case: Apply the same method as in the previous section, considering that d M and allconsidered measures are σ -invariant and that any CA is Lipschitz w.r.t d M . (cid:3) .6. Extension to other cellular automata Definition 14. Let F , F be two CAs on A Z and B Z , respectively. We say that F factorises onto F if there exists a factor π : A Z → B Z such that π ◦ F = F ◦ π .In particular, if F admits a particle system ( P , π , φ ) , then F admits a particle system with ( P , π ◦ π , φ ) .In this section, we extend the Theorems 4 and 6 to automata that factorise onto a gliders au-tomaton, and discuss conditions for the extension of Theorem 7. In Section 2.5, we exhibited ageneral method to find such a factor using experimental intuition when such a factor is not obvious.In other words, we extend our results to automata that admit a particle system ( P , π, φ ) , where P = {− , +1 } and φ updates the particle positions similarly to a gliders automaton.In order to extend the theorem to such CAs, starting from an initial measure µ , we must firstensure that π ∗ µ ∈ M ix . We show that the third condition in the definition of M ix is invariantunder morphism. Proposition 10. Let π : A Z → B Z be a morphism, µ ∈ M σ ( A Z ) and k > any real such that (cid:80) n ≥ α µ ( n ) k < ∞ . Then, (cid:80) n ≥ α π ∗ µ ( n ) k < ∞ .Proof. We keep the notations from the definition of α µ ( n ) . π is defined by a local rule with neigh-bourhood N ⊂ [ − r, r ] for some r > . Then, π − B ] −∞ , ⊂ B ] −∞ ,r ] and π − B [ n, + ∞ [ ⊂ B [ n − r, + ∞ [ .By σ -invariance, we have for all n α π ∗ µ ( n ) < α µ ( n − r ) , and the result follows. (cid:3) Hence, if µ ∈ M ix , we only have to prove that π ∗ µ weighs evenly the sets of particles − and +1 , and that the corresponding asymptotic variance is not zero. Under these assumptions, we canextend some of the previous results with the forbidden patterns playing the role of the particles. Corollary 4. Let F : A Z → A Z be a CA and µ ∈ M σ ( A Z ) . Suppose that F factorises onto a ( v − , v + ) -GA via a factor π such that π ∗ µ ∈ M ix .Then Theorem 4 and the first point of Theorem 6 hold if we replace “ x k = ± ” by “ π ( x ) k = ± ”. Even if µ is a simple, e.g. Bernoulli measure, π ∗ µ can fail to satisfy the first and second conditionof M ix . We provide a counterexample at the end of this section. Examples:Traffic automaton: Let A = { , } and F be the elementary CA corresponding to rule as defined in Section 2.6.1. F factorises on the ( − , +1) -gliders automaton, usingthe factor introduced in that section: (cid:55)→ +111 (cid:55)→ − , (cid:55)→ This factor is represented in Figure 5. If µ is a measure such that π ∗ µ ∈ M ix , thenTheorem 4 and the first point of Theorem 6 hold.For example, this is true for the 2-step Markov measure defined by the matrix (cid:18) p − p − p p (cid:19) and the eigenvector (cid:18) / / (cid:19) with p > . A particular case is the Bernoulli measure of pa-rameters ( , ) .The upper bound in Theorem 7 can also be extended by considering the fact that d M ( F t ∗ µ, (cid:99) δ ) ≤ (cid:88) n ∈ N n π ∗ F t ∗ µ (cid:16) A Z \ [0 n ] (cid:17) . igure 14. The 3-state cyclic CA, a one-sided captive CA and the product CA.From there the upper bound can be obtained as in the original proof. Let A = Z / Z and C be the 3-state cyclic automaton. Weconsider the factor π defined in Section 2.6: ab (cid:55)→ +1 if a = b + 1 mod 3 ab (cid:55)→ − if a = b − ab (cid:55)→ if a = b If µ is such that π ∗ µ ∈ M ix , then Theorem 6 applies. This is true in particular when µ is any 2-step Markov measure defined by a matrix ( p ij ) ≤ i,j ≤ satisfying p + p + p = p + p + p , all of these values being nonzero, with ( µ i ) ≤ i ≤ its only eigenvector. Thisincludes any nondegenerate Bernoulli measure. However, even when the limit measure isknown (e.g. starting from the uniform measure), Theorem 7 does not apply directly. One-sided captive automata: Let F be any one-sided captive cellular automaton definedby a local rule f . As explained in Section 2.6, F factorises onto the ( − , -gliders automatonwith a factor defined by: ab (cid:55)→ +1 if a (cid:54) = b, f ( a, b ) = aab (cid:55)→ − if a (cid:54) = b, f ( a, b ) = bab (cid:55)→ if a = b For an initial measure µ , if π ∗ µ ∈ M ix , then Theorem 4 and the first point of Theorem 6apply.Notice that this class of automata contains the identity ( ∀ a, b ∈ A , f ( a, b ) = b ) and theshift σ ( ∀ a, b ∈ A , f ( a, b ) = a ). However, since we have in each case π − (+1) = ∅ or π − ( − 1) = ∅ , it is impossible to find an initial measure that weighs evenly each kind ofparticle, and so π ∗ µ cannot belong in M ix . The limit measure, however, depends on theexact rule, and Theorem 7 does not apply directly. Counter-example:Product automaton: Let A = Z / Z and F be the CA of neighbourhood {− , , } definedby the local rule f ( x − , x , x ) = x − · x · x . Using the formalism from Section 2.5, we can ee that F factorises onto the ( − , -GA by the factor π : → +110 → − otherwise → If µ is any Bernoulli measure, then π ∗ µ satisfies all conditions of M ix except that σ µ =0 ; indeed, we can check that for π ∗ µ -almost all configurations, the particles +1 and − alternate. Hence, only one particle can cross any given column after time 0, and therefore ∀ α > , µ (cid:16) T − n ( x ) n ≤ α (cid:17) −→ n →∞ . Furthermore, any particle survives up to time t only if itis the border of a initial cluster of black cells larger than t cells, which happens with aprobability µ ([1]) t decreasing exponentially in t .Even though we showed that the asymptotic distributions of entry times are known for some classof cellular automata and a large class of measures, this covers only very specific dynamics. 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E-mail address : [email protected] URL : http://mat-unab.cl/~hellouin/ Aix Marseille Université, CNRS, Centrale Marseille, I2M UMR 7373, 13453, Marseille, France E-mail address : [email protected] URL :