PERCOLATION TIMES IN TWO{DIMENSIONAL MODELS FOR EXCITABLE MEDIA

. The three-color Greenberg{Hastings model (GHM) is a simple cellular automaton model for an excitable medium. Each site on the lattice Z 2 is initially assigned one of the states 0, 1 or 2. At each tick of a discrete{time clock, the con(cid:12)guration changes according to the following synchronous rule: changes 1 ! 2 and 2 ! 0 are automatic, while an x in state 0 may either stay in the same state or change to 1, the latter possibility occurring i(cid:11) there is at least one representative of state 1 in the local neighborhood of x . Starting from a product measure with just 1’s and 0’s such dynamics quickly die out (turn into 0’s), but not before 1’s manage to form in(cid:12)nite connected sets. A very precise description of this \transient percolation" phenomenon can be obtained when the neighborhood of x consists of 8 nearest points, the case (cid:12)rst investigated by S. Fraser and R. Kapral. In addition, (cid:12)rst percolation times for related monotone models are addressed.


Introduction.
The Greenberg-Hastings model (GHM) ( [WR], [GH]) on Z 2 tries to capture two essential properties of a two-dimensional excitable medium: a refractory phase and excitation by contact. In one of its simplest incarnations, GHM dynamics have two parameters: an integer number κ ≥ 3, which we refer to as the number of colors (or states), and a finite set N ⊂ Z 2 , containing the origin, which is called the neighborhood . The neighborhood of an arbitrary site x ∈ Z 2 is then obtained by translation: N x = x + N . The state space of the GHM is {0, 1, . . ., κ − 1} Z 2 and the update rule is given as follows: (1.1) γ t+1 (x) = (γ t (x) + 1) mod κ, if γ t (x) > 0 or there is a y ∈ N x with γ t (y) = 1, 0, otherwise.
In words, all colors 1, . . . , κ−1 update automatically to 2, . . ., 0, respectively, while 0 updates by contact with a least one color 1 in its neighborhood. Readers interested in mathematical analysis of this GHM rule and various generalizations may consult [DS], [FGG1], [FGG2], [DN], [Dur], [DG], and [Ste]. Researchers closer to applications have considered many classes of multiparameter rules related to the GHM (see, for example, [WTW] and the references contained therein).
Assume for a moment that N is the five point set {(0, 0), (±1, 0), (0, ±1)}, the nearest neighbor case. Assume moreover that γ 0 is a product measure with P (γ 0 (x) = 0) > 0, P (γ 0 (x) = 1) > 0 and P (γ 0 (x) = i) > 0 for some i ∈ {2, . . ., κ − 1}. Then it turns out that the dynamics is eventually run by "clocks," i.e. objects like which repeat their pattern every κ units of time and eventually cause other sites to do the same, giving rise to local periodicity (see [FGG2]). The main aim of this paper is to study the case when γ 0 contains only 0's and 1's, which results in a markedly different dynamics. To see why, start a nearest neighbor GHM with just a single 1 in a sea of 0's. This initial state generates a diamond-shaped ring which keeps expanding forever. If the initial state contains just two 1's, the two rings proceed to annihilate on the boundary and combine into one big ring, which again expands without bound. We will show that such annihilating rings are the dominant feature of the dynamics started from a product measure consisting only of 0's and 1's. Indeed, one inevitably reaches such a conclusion by doing just a few computer simulations.
As in the simple example above, the number of colors turns out to be completely immaterial for the ring dynamics. Thus we assume that κ = 3 from now on. We also assume that the initial state γ 0 is the product measure given by P (γ 0 (x) = 1) = p = 1 − P (γ 0 (x) = 0) for every x. Since our main results involve various percolation properties of subsets of Z 2 , we must start with a few definitions.
Fix a finite set D ⊂ Z 2 . We say that a set G ⊂ Z 2 D-percolates if there exists an infinite sequence x 1 , x 2 , . . . of distinct sites in G such that x k+1 − x k ∈ D for k = 1, 2, . . .. The usual site percolation, which we call 1 -percolation, is obtained by taking D = B 1 (0, 1), while its dual, which we refer to as ∞ -percolation, is given by D = B ∞ (0, 1). (We adopt the usual notation for balls: The study of ring growth and interaction in this simple model of an excitable medium fits into the general framework of investigating domain growth in physical systems: namely, how spatial structures arise far from equilibrium. Real-life phenomena that may be related to the ones discussed here include aspects of the Belousov-Zhabotinsky reaction and processes in cardiac tissue ( [FK], [Kap], [KW], [SLC]). Indeed, ring dynamics may be the dominant feature of conduction of electrical impulses in a healthy heart, while spatial defects (e.g. scar tissue) can break the rings and cause, through the formation of clock-like excitation centers, atrial fibrillation. These natural phenomena were the main motivation of S. Fraser and R. Kapral, who studied ring dynamics in excitable media in [FK]. They chose the Moore neighborhood (i.e. N = B ∞ (0, 1)) and combined computer simulations with various intuitively compelling but non-rigorous arguments, to arrive at the following three conclusions: (1) Each point is in state 1 exactly once.
(2) The density of 1's goes to 0 very quickly as time t increases, and is of order √ p at its maximum.
(3) There exists a narrow time interval, lasting at most two time units, during which 1's percolate. ( [FK] is vague as to what type of percolation this is.) Assuming (1), (2) follows readily, as shown in [Bra]. But (1) is not very hard to show either (see Corollary 2.2). The most interesting statement is of course (3); our main aim in the present paper is to give a rigorous and more detailed treatment of this issue.
Before we proceed, let us introduce some standard regularity assumptions on the neighborhood N which arose in our previous work ( [GG]). We call N symmetric if −N = N and supercritical if the origin is an interior point of the convex hull co(N ). Moreover, denote the sum N + N + · · · + N of n copies of N by N n (with the conventions N 0 = {0}, N n = ∅ when n < 0), and call N irreducible if ∪ n≥1 N n = Z 2 . Note that irreducibility implies supercriticality, but not vice versa. We assume throughout this paper that N is both symmetric and irreducible.
Theorem 1. There exist sufficiently large integers a and b (depending only on N ) so that: (1) For every p > 0 there exists a time T (p) at which {γ T (p) = 1} B ∞ (0, a)-percolates.
(3) T (p) √ p converges to a finite non-zero constant as p → 0.
To prove a more precise theorem, we restrict to the special case of Moore neighborhood considered in [FK].
(2) There is a countable set S ⊂ (0, 1) with 0 as its only limit point, such that the following is true. Let I 1 , I 2 , I 3 , . . . be the successive intervals in (0, 1] \ S, where I 1 is the one that includes 1. Let G 1 be the union I 1 ∪ I 3 ∪ I 5 ∪ . . . and G 2 = (0, 1] \ S \ G 1 . If p ∈ G 1 , then 1's in γ t 1 -percolate at t = T (p) but do not ∞ -percolate for t = T (p). If p ∈ G 2 , then 1's in γ t never 1 -percolate, but they ∞ -percolate for both t = T (p) and t = T (p) + 1. Finally, if p ∈ S, then 1's in γ t ∞ -percolate only for t = T (p) and at this time they do not 1 -percolate. , and 5 and 6 (bottom). At times 4 and 6 (resp. at time 5) the sites with color 1 that are connected via an l ∞ -path (resp. an l 1 -path) to the left edge of the rectangle are painted black. The remaining 1's are painted dark gray, while 2's are painted light gray. It is clear that at time 3 the rings of 1's are still too small to be able to ∞ -percolate. This remains true at time 4, despite the existence of relatively long ∞ -paths.
At time 4, 1's form a left-right 1 -connection of the square, suggesting 1 -percolation. Finally, by time 5 there is enough annihilation between the rings so that 1's cannot even ∞ -percolate. These figures leave little doubt that T (0.01) = 5 and 0.01 ∈ G 1 .
It is natural to ask what happens to Theorems 1 and 2 in more general circumstances. One possible generalization is to consider an arbitrary excitation threshold θ; this means that x will change color from 0 to 1 iff the number of sites with color 1 in N x exceeds θ ( [FGG1], [FGG2], [DG], [GG]). In this case "failed centers" can break up the rings and cause local periodicity; computer simulations show that this is indeed a likely scenario in many regimes, though a rigorous general statement is hard to prove. Another interesting class of questions concerns the expanding "shells" in dimensions 3 or higher. These higher dimensional models are considerably harder to deal with not only because of technical difficulties, but also because of qualitatively different "intermediate" regimes. On a different note, present methods can be extended to handle various related two-dimensional models with recurrent ring dynamics ( [Gra3]). The first step in the proof of the above theorems is a comparison between the GHM and a monotone 2-color model which is in fact the threshold 1 (often called additive) version of threshold growth dynamics ( [GG]). This auxiliary monotone automaton with state space {0, 1} Z 2 is denoted by ξ t and given by the update rule: Thus, {ξ t = 1} = {ξ 0 = 1} + N t . Assume that ξ 0 . like γ 0 , is a product measure of 0's and 1's, with a density of 1's given by p > 0. Define T a (p) to be the first time for which the set {ξ t = 1} B ∞ (0, a)-percolates. By the ergodic theorem, T a is a deterministic function of p. It is also not hard to prove the following result.
Theorem 3. As p → 0, T a (p) √ p converges to a finite non-zero constant λ c , which depends on N but is independent of a.
Many models for crystal formation and growth, such as the famous Johnson-Mehl model ( [JM]), include continuous seeding of nuclei. This inspired R. Kapral and M. Weinberg ([KW]) to propose a variant of ξ t . This model, denoted byξ t , is a Markov chain with state space {0, 1} Z d and the following transition rule at every site x: Hence this rule is a two-stage procedure: at the first stage the dynamics of ξ t are applied to a configuration of 0's and 1's, and at the second stage a proportion p of the remaining 0's are turned to 1's. Startξ t from the quiescent state:ξ 0 (x) = 0 for all x. Now defineT a (p) to be the first time at which {ξ t = 1} B ∞ (0, a)-percolates. Our next theorem identifies the behavior ofT a near p = 0. (We will see later thatT a is a deterministic function of p as well.) Theorem 4. As p → 0,T a (p)p 1/3 converges to a finite non-zero constant. This theorem is illustrated by Figure 2, which shows the state of the 500 × 375 system with N = B ∞ (0, 1) and p = 0.0005 at time t = 12, the first time when there is an ∞ -path connecting left and right edges of the rectangle. The picture suggests thatT 1 (0.0005) = 12.
The rest of the paper is organized as follows. Section 2 contains a few results on additive growth, and a proof of Theorem 3. Theorem 4 is proved in Section 3. Section 4 states a few well known results about long-range percolation models, which we then use in Section 5 to prove Theorem 2. Finally, Section 6 completes the proof of Theorem 1.

Growth dynamics with initial nucleation.
We start with a simple lemma which establishes a connection between the growth automaton ξ t and the GHM γ t .
Proof. Denote the set on the right of (2.1) by A t . To prove (2.1) by induction, observe first that it is true at t = 0, and then assume that {γ s+1 = 1} = A s for all s < t.
To estimate the maximal value this expression can have, we first use the shape theorem; this result was first proved by Willson ([Wil]), but we give a short proof in Lemma 2.3 below. This theorem implies that |N n |/n 2 → Area(co(N )) as n → ∞. (Here, and in the sequel, we denote by | · | the cardinality and by Area the Lebesgue measure.) Assume, for a moment, that the growth at the boundary is also regular enough so that for some constant c > 0. Then c = 2Area(co(N )), and a short computation gives In fact, we show below (Lemma 2.4) that (2.2) is always true. Essentially, we show that the boundary of N n is quickly trapped into a stable pattern. Figure 3 provides an illustration in which points in N n \ N n−1 are periodically shaded.
Proof. Take m ≥ 2 and integers x 1 < x 2 < · · · < x m and M = {x 1 , . . ., x m }. If a = g.c.d.(M − x 1 ), then elementary number theory implies that there exists a constant M 0 so that Let e 1 , e 2 be the usual basis vectors and assume that β M is such that N ⊂ Ze 1 + (−∞, β M ]e 2 . Moreover, assume that N l = min{α : αe 1 +β M e 2 ∈ N } and N r = max{α : αe 1 +β M e 2 ∈ N } are different, N l < N r . Then, we claim that, for any fixed k ∈ {0, 1, 2, . . .}, either (Ze 1 + n(β M − k)e 2 ) ∩ N n = ∅ for all n, or else there exists positive integer constants m k , M k and a Indeed, for k = 0, (2.4) follows immediately from (2.3). In words, we know that the last horizontal line of N n in the vertical direction is periodic, away from the edges. But then this argument shows that the same must be true for the next furthest line, and so on. Hence an induction argument with a repeated application of (2.3) establishes (2.4). By Lemma 2.3, for large enough k, m k = 1, a (k) 1 = 1, and we can take M k = M . This means that at least the boundary in the e 2 direction stabilizes away from the edges.
However, a similar argument (with a few more notational complications) will work in any direction u such that a line with normal u contains at least two extreme points of co(N ). Only finitely many such directions are determined by the boundary of co(N ) and it is easy to combine them to end the proof.
Next up is a well-known result about a continuous percolation model (for a proof, see [ZS] and [Roy]), which identifies the constant λ c in the statement of Theorem 3.
In this section, we only need the case when all the U x are point masses at 1. To prove Theorem 3, we use a basic coupling between W (λ) and ξ t which is similar to the one used by Zuev and Sidorenko in their pioneering papers ( [ZS]).
Proof of Theorem 3. Fix the density p and define Q(x, p) = {y ∈ R 2 : || √ px − y|| ∞ ≤ √ p/2}. Let P be a Poisson random field with density 1. Declare a site x ∈ Z 2 to be occupied if Q(x, p) ∩ P = ∅. Since Area(Q(x, p)) = p, and Area(Q(x, p) ∩ Q(y, p)) = 0 for x = y, the sites in Z 2 are independently occupied, each with probability 1 − e −p . Define ξ 0 by making {ξ 0 = 1} equal to the set of occupied sites.
We claim that there exists a constant N > 0 (again dependent only on N ) so that, for every t and p.
To prove (2.5), assume first that a point x ∈ R 2 is in the leftmost set of (2.5). Then there is a y ∈ P such that x ∈ y + (t − N ) √ p · co(N ). Find a z ∈ Z 2 so that ||y − √ pz|| ∞ ≤ √ p/2. This means that ξ 0 (z) = 1 and hence the middle set in (2.5) covers √ pz + √ p(t − M ) · co(N ) by Lemma 3.3. Since N is supercritical, B ∞ (0, 1/2) ⊂ co(N ), and so x is in the middle set of (2.3) if we choose N = M + 1.
It is easy to translate (2.5) into a statement about T a . Namely, , finishing the proof.

Growth dynamics with continuous nucleation.
We start the analysis ofξ t by associating to every site x, independently of other sites, a geometric random variable S x with P (S x = k) = p(1 − p) k−1 , k = 1, 2, . . .. Then, Note thatT a (p) is a translation invariant function of the independent family {S x : x ∈ Z 2 }, hence a determinstic quantity. Moreover, (3.1) and (2.2) yield, after a straightforward computation, If we denote the first passage time to the origin by τ , i.e. τ = min{t : ξ t (0) = 1}, then (3.2) immediately implies that p 1/3 τ converges weakly as p → 0 to a non-degenerate random variable. Therefore, most sites make the 0 to 1 transition at times of the order p 1/3 , which certainly makes Theorem 4 plausible.
Lemma 3.1. If λ < λ c then there exists an > 0 so that the -fattening W (λ) does not percolate.
where U x are now independent and uniform on [a, b] and P(µ) is a Poisson point location with intensity µ. Note that W (λ) = W ([ , λ + ], 1).
Let R t (x) equal 0 if S x ≥ t and t − S x otherwise. This is the radius of the growing set which was generated by a seed at x during the time interval [0, t]. Observe that P (R t+1 (x) = r | R t+1 (x) > 0) = u r , r = 1, . . . , t, where u r are defined in the statement of Lemma 3.3. Now, assume that P is an intensity 1 Poisson random field and declare a site This makes sites open independently with probability P (R t+1 (x) > 0). Adjoin to each site x that is open at time t a random variable V t+1 (x) with P (V t+1 (x) = r) = u r , r = 1, . . . , t, independently of other points. We get, by (3.1), Assume that the claim of the theorem is false. Then, we can find some sequence p n converging to 0, and λ ∈ [0, ∞] \ {λ c }, such thatT a (p n )p 1/3 n → λ 2/3 as n → ∞. We will assume that λ ∈ (0, ∞); the other two cases are then treated easily.

Some percolation preliminaries.
In this section, we state the results needed to prove Theorem 2. These are either applications of general percolation theory, or else follow by minor modifications of well-known arguments, so detailed proofs will be omitted.
Of course, almost exactly the same arguments handle the ∞ -percolation case: Lemma 4.2.
(1) If p ≤p k c , then there exists an infinite sequence of disjoint k-vacant 1 -circuits.
(2) If p ≥p k c , then there exists an infinite sequence of disjoint k-occupied ∞ -circuits.
(3) The k-occupied sites ∞ -percolate iff p >p k c , while k-vacant sites 1 -percolate iff p <p k c .
It is obvious that p k+1 ≤p k ≤ p k for all k ≥ 0. For many of our arguments it is however crucial that strict inequalities hold.
Proof. To apply the arguments from Section 3 of [AG], note that 1 -percolation (resp. ∞percolation) of k-occupied sites is equivalent to the site percolation on the graph with vertices Z 2 and edges between any two sites x and y for which y − x ∈ B ∞ (0, 2k) + B 1 (0, 1) (resp. y − x ∈ B ∞ (0, 2k + 1)).

5.
Ring dynamics in the GHM: the Moore neighborhood case.
We begin with some notation. By dist r we mean distance taken in the norm ||·|| r . Choose any set A ⊂ Z d . Define its ∞ -boundary ∂ ∞ A to consist of all sites x ∈ A for which dist ∞ (x, A c ) = 1, and the outside ∞ -boundary by ∂ ∞ o A = ∂ ∞ (A c ). Note that Lemma 2.1 then says that The proof of Theorem 2 starts with a geometrical lemma which is probably well-known.
. Let x and y be connected by an 1 -path (resp. ∞ -path) contained in A c and connected by an 1 -path (resp. ∞ -path) contained in A ∪ ∂ ∞ o A. Then x and y can be connected by an 1 -path (resp. ∞ -path) contained in ∂ ∞ o A.

Proof.
A path (an 1 -path or an ∞ -path) will be called self-avoiding if the induced path in R 2 (obtained by adding line segments between successive points) does not cross itself. Moreover, for two self-avoiding paths with the same endpoints π 1 and π 2 , we define K(π 1 , π 2 ) to be the subset of R 2 consisting of the union of the two induced paths together with all bounded components of the complement of this union.
We start by proving the 1 version. Let π 1 be a self-avoiding 1 -path connecting x to y through A ∪ ∂ ∞ o A and let π 2 be a self-avoiding 1 -path connecting x to y through A c .
If π 1 and π 2 do meet at sites other than x and y, let x = v 0 , v 1 , . . . , v n = y be all points of intersection. Then do the above for x replaced by v k and y replaced by v k+1 ; k = 0, . . . , n − 1.
Finally, we prove the ∞ version. Let π 1 be a self-avoiding l ∞ -path connecting x and y through A ∪ ∂ ∞ o A, and let π 2 be a self-avoiding l ∞ -path which connects x and y through A c . Then ρ is defined in a way analogous to the 1 case. Again, to prove that ρ consists of sites on ∂ ∞ o A, one needs to define a transformation that will eliminate z / ∈ ∂ ∞ o A from ρ. The transformation is given by Figure 4 (all other cases are rotations or reflections of those drawn). Note that every new point on ρ is within 1 -distance 1 of z, and, because z is not on π 1 , this implies that new points cannot be on the right of π 1 . Apart from this issue, the argument is the same as before.
Proof of Theorem 2. Start by defining S = ∪ k≥0 {p k c ,p k c }.
Assume first thatp k c < p < p k c for some k ≥ 0. Then {ξ t = 1} ∞ -percolates at t = k, but neither 1 -percolates for t ≤ k nor ∞ -percolates for t < k. We first prove that {γ t = 1} ∞ -percolates for t = k. This is obvious for k = 0, so we can assume that k ≥ 1.
They cannot ∞ -percolate in γ t for t ≥ k + 2 since 0's in ξ k+1 do not ∞ -percolate (Lemma 4.1 (3)). This proves the claim of the theorem, with T (p) = k + 1 in this case.
6. Ring dynamics in the GHM: the general neighborhood case.
Proof of Theorem 1. Throughout this proof C is a generic positive integer constant, which may change value from one appearance to another, and depends only on N . Recall the coupling of ξ t and W from the proof of Theorem 3 which establishes (2.5). That proof and Lemma 2.5 then imply that there exists a C so that if t 2 > λ c p −1/2 + C, then {ξ t 2 = 1} 1 -percolates and the infinite 1 -cluster contains an infinite sequence of disjoint 1 -circuits. On the other hand, if t 1 < λ c p −1/2 − C, then {ξ t 1 = 0} ∞ -percolates.
However, if a is large enough, then {x k } forms an infinite B ∞ (0, a)-percolating set, proving part (1).
Parts (2) and (3) follow easily from Theorem 3 and its proof.