FOUR SNAPSHOTS. 1. Introduction

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1 SELF-ORGANIZATION OF RANDOM CELLULAR AUTOMATA: FOUR SNAPSHOTS DAVID GRIFFEATH Department of Mathematics University of Wisconsin Madison, WI U.S.A. Abstract. We discuss four very simple random cellular automaton (CA) systems that self-organize over time. The first is a directed interface process which stabilizes in a coherent statistical equilibrium. The second is a model for excitable media: nucleating spiral cores lead to a locally periodic final state. The third model is a prototype for curvature-driven clustering. And the fourth illustrates the evolution of complex viable structures near phase boundaries in a parameterized family of non-linear population dynamics. For each CA we present a mix of rigorous results, conjectures, and empirical findings based on computer experimentation. Key words: Cellular automaton, interacting particle system, interface, excitable medium, selforganization, nucleation, metastability, artificial life. 1. Introduction By a cellular automaton (CA) we mean a spatially-distributed dynamical system that evolves via local, homogeneous, parallel updating. Somewhat informally, we will call a CA random if its evolution has random ingredients, either in the starting state or in the dynamics. Deterministic CA rules may be viewed as digital counterparts of partial differential equations. Like their more traditional relatives, they can emulate a broad range of fundamental spatio-temporal phenomena across the spectrum of applied science. Random CA models include discrete-time variants of the interacting particle systems that are a mainstay of mathematical physics; the synchronous CA versions are ideally suited for real-time simulation on parallel computing devices. See [TM] for a nice, practical introduction to CA algorithms. Rick Durrett s St. Flour lectures [Dur2] survey many recent rigorous results for interacting random systems, with emphasis on co-existence of phases and connections with partial differential equations. Our goal here is to describe four kinds of random CA that serve as prototypes for various non-linear complex systems. Each example is simple enough that a substantive, rigorous mathematical analysis seems within reach. The unifying theme is self-organization: a tendency toward large-scale, coherent structure starting from disordered initial states. We will focus on irreversible dynamics somewhat beyond the purview of traditional statistical mechanics. Rather, our models are motivated by problems from computer science, chemistry, and biology. Theoretical researchers in those fields are beginning to use random CA models to gain insights into organizational, adaptive, and evolutionary principles of spatially-distributed dynamics.

2 50 DAVID GRIFFEATH Their ideas constitute a rich new source of important problems in stochastic processes, while probability theory, in turn, has much to contribute to their efforts. 2. Asynchronous Deterministic Computation: A Directed Interface Our first snapshot comes from computer science [Tof]. Imagine a two-dimensional integer array of cpu s, with nearest neighbor connections, each assigned to carry out a sequence of calculations. In order for the machine at x to perform its (n + 1) th job reliably, it must wait until all neighboring machines have completed their n th jobs (in order to access needed information from the network). Typically, the times required to complete the various jobs are rather unpredictable, so an organizational algorithm is required to keep the cpu s in synch. One can accomplish this by attaching to each node in the network a phase variable that keeps track of how many jobs have been completed, and prohibits a cpu from proceeding with its computation whenever its phase is ahead of any of the phases of its neighbors. In practice, the phases need only cycle through four states: 0, 1, 2, 3, say. If all sites are initially in phase, then the variables at neighboring nodes will never differ by more than one in modulus, and so four states are sufficient to determine whether a neighbor of x is ahead, equal to, or behind the machine at x. Of course the waiting protocol slows down each individual cpu, but only by a constant factor independent of the size of the network. In many contexts this is a small price to pay for the benefits of parallel computation. Toffoli and Margolus [TM] have proposed the following random CA prototype for such a synchronization scheme. Assume that the durations of jobs are i.i.d. geometric with parameter p. Imagine a directed interface representation of the dynamics on the space time lattice Z 2 N, where the last coordinate codes the number of jobs completed. Write ξ t (x) = n to signify that the machine at x has completed n jobs at time t. Then starting from the flat state ξ 0 (x) 0, at all times t the system consists of a connected interface, by which we mean that ξ t (x + 1) ξ t (x) {0, 1} for all x, t. The discrete-time dynamics are captured by the following maxim. Relative minima advance one unit with probability p. (1) See page 95 of [TM] for pictures. Observe that the shape of the interface in any finite network of this sort (i.e., the state modulo translations in n) comprises an irreducible aperiodic finite Markov chain. Thus the shape chain converges to a unique equilibrium, and so the interface advances at an asymptotic speed given by the invariant density of relative minima. But this stationary measure appears intractable, so the finite state space theory is of little use in the analysis of large networks. Rather, one wants to know that the directed interface ξ t (x), defined on all of Z 2 N, advances with a positive asymptotic speed α = α(p), and that the infinite system is well-approximated by large finite ones. We recently learned an elegant approach to the speed question for the directed interface process from Harry Kesten. The idea is to establish an exact connection with last passage percolation (cf. [GK]). Attach to each (x, n) a random variable X (x,n) that records how long the n th job at x takes. (This notation works out best if the initial job is considered the 0 th.) Let T (x,n) denote the time until the n th job at

3 SELF-ORGANIZATION OF RANDOM CELLULAR AUTOMATA 51 x begins. We claim that the distribution of T (x,n) is given by the maximum of the n summands X over all paths connecting Z 2 {0} to (x, n) in the directed graph (Z 2 N, E), where E consists of edges from each site (x, m) to all (y, m 1) such that y is a neighbor of x. Note that in the two-dimensional nearest-neighbor case the max is over 5 n paths, and that the n variables on each path range from level 0 to level n 1. The proof of the claim is a straightforward induction on n. In essence, T (x,n) equals the maximum of T (y,n 1) over any of the neighbors y of x (including x itself) plus the times it takes to carry out the (n 1) th job at x. Using this last passage percolation representation, and denoting the origin of Z 2 by 0, one can show that T (0, ) is superadditive. Hence, by the Kingman Liggett Ergodic Theorem (see Dur1] or [Lig]), 1 n T (0,n) γ as n a.s. and in L 1. Straightforward inversion then yields a limiting speed for the interface: 1 t ξ t (0) α = γ 1 as t a.s. (2) The gist of the superadditivity property is to show that T (0,m+n) T m + T n, (3) where T m and T n are independent, with the same distributions as T (0,m) and T (0,n), respectively. Let us sketch the derivation of (3). Starting at (x, m + n), trace back the last passage path 1 to Z 2 {m}, let T n denote the time along this path, and let y be its final position. Then, starting at (y, m), trace back the last passage path 2 to Z 2 {0}, and let T m denote the time along this path. For any value of y, T n has the same distribution as T (0,n) by translation invariance. Moreover, it is clear that T (0,m+n) must be at least as large as the sum of variables X along the concatenated path 1 2, as desired. Of course, to know that our network doesn t grind to a halt, we need to ensure that α > 0, or equivalently that γ <. The last percolation representation facilitates a Peierls style argument to this effect. Namely, since T (x,n) is a maximum over 5 n sums of n independent geometric random variables with parameter p, P (T (x,n) > a) 5 n P (S n > a). If a > 6/p the right side tends to 0 at an exponential rate in n, by a standard large deviations computation. Thus Borel Cantelli implies that α(p) > p 6. Superadditivity gives little insight into the stability of the directed interface however. Indeed, it seems challenging to extend the above analysis to initial configurations with non-zero asymptotic slope. To dig deeper, a more illuminating technique is coupling. Let us discuss this approach in the one-dimensional setting, where it is most effective. Here the interface may be depicted as a polygonal function f : Z N

4 52 DAVID GRIFFEATH or, by extension, f : Z Z, with nearest neighbor edges of three types (moving left to right):,, and. Adopting the usual orientation, relative minima of this polygonal function move up in parallel, each with probability p. We can imagine starting the directed interface ξ t from the flat state ξ 0 (x) 0, or from any other polygonal f. The basic coupling gives a way to represent two directed interfaces on the same probability space, and on the same space time diagram Z Z: whenever both interfaces have a relative minimum at a site x, they use the same probability p coin toss to decide whether to move up. By restricting the state space to a finite one-dimensional lattice with wrap-around edges, it is a simple matter to simulate the basic coupling of ξ t on a computer. We carried out many such simulations several years ago. For instance, Figure 1 shows the simultaneous evolution, with p = 1 2, of the flat initial state ( ) and an initial state of maximal oscillation ( ) on a periodic array of 1,000 sites, at time t = 1,000 and then t = 10,000. The flat state rapidly settles into a stable equilibrium. The state undergoes a much slower (hydrodynamic) transition to the same equilibrium since its peak cannot equilibrate until stochastic effects propagate from the valley. What is clear from the simulation is that the two interfaces are very successfully coupled in a neighborhood of the original minimal site of the state. By this we mean that the great majority of pairs of edges have the same slopes ( 1, 0, or 1) at the same locations in that neighborhood, so the difference between the interfaces is almost constant there. Evidently, as time goes on, the region of successful coupling grows and the density of discrepancies tends to 0. This is strong empirical evidence for loss of memory in the directed interface process, and hence for the existence of a unique equilibrium starting from any profile with asymptotic slope 0. We should note that if f has non-zero asymptotic slope m, then this slope will be preserved by the dynamics, and hence such an f cannot possibly be successfully coupled to the flat state. Rather there should be a unique equilibrium for each slope in m [0, 1), each with its own characteristic speed α m (p). It also seems clear that slope 0 interfaces should propagate most rapidly, and that α m (p) 0 as m 1, since a slope 1 interface cannot move at all. Of course simulations provide little understanding as to why a coupling works. Quite recently Larry Gray [Gra] has established the stability of the one-dimensional directed interface process by proving that the basic coupling is successful. The strong law (2) follows as a corollary of his equilibrium analysis, which applies to any initial configuration with an asymptotic slope m. In essence, Gray s method exploits a Lyapunov function for the coupled increment processes. Since each increment of the interface has slope 1, 0, or 1, the difference between coupled slopes at any location is an integer in [ 2, 2]. Thinking of these discrepancies as signed particles, at most two per site, one can check a key monotonicity property (which, unfortunately, does not extend to dimensions d 2). Namely, particles of opposite sign can annihilate, and particles can move to neighboring locations, but new particles are never created. Using this observation, and the fact that annihilations must inevitably occur sooner or later, Gray proves that the density of coupling discrepancies tends to 0 starting from any two configurations with the same m. He is then able to extend techniques of Liggett [Lig], developed originally for exclusion processes, to conclude that there is a unique extremal invariant measure π m = π m (p) for each slope m, and that any

5 SELF-ORGANIZATION OF RANDOM CELLULAR AUTOMATA 53 Fig. 1. Basic coupling of the directed interface. initial interface with slope m settles down to π m. The same techniques apply to a fairly broad class of one-dimensional models that can include, for instance, advances of more than one unit as well as retreats. Random interface dynamics are notoriously difficult to analyze rigorously since their conserved quantities tend to give rise to self-organized distributions with longrange correlations. A few such processes are isomorphic to simple exclusion models. For example, discrete-time one-sided exclusion on Z is equivalent to an interface of and increments with update rule (1) except that the advance is two units. In large part because product measures are invariant for simple exclusion, there is an amazingly rich theory available in this case, and even more detailed results continue to appear. However most interface equilibria are computationally intractable, in which case robust methods are needed. Last passage percolation and basic coupling now seem to provide the beginnings of a more general theory. 3. Excitable Cellular Automata: Nucleating Waves Mathematical models for excitable media attempt to capture and explain the key features of periodic wave transmission through environments such as a network or tissue. Since the pioneering work of Wiener and Rosenblueth [WR], a great many researchers from the applied sciences have adopted various modeling frameworks,

6 54 DAVID GRIFFEATH most notably partial differential equations, cellular automata, and coupled lattice maps. A common feature of many of these models is the requirement that some threshold level of excitation occur in a neighborhood of a location in order for that location to become excited and conduct a pulse. Such activity is typically followed by a refractory period in which further excitation is inhibited. Physical systems that exhibit this basic phenomenology include neural networks, cardiac muscle, and the Belousov Zhabotinsky (BZ) oscillating chemical reaction. In two dimensions, excitable systems are typically characterized by the emergence of spatially-distributed stable target patterns or spirals. With the advent of effective computer visualization technology there has been a recent flurry of excitable medium modeling that tries to approximate precise quantitative features of observed phenomena (e.g., curvature and wave velocity in the BZ reaction). To accomplish this, most experimentalists introduce several rather ad hoc parameters designed to generate an assortment of non-linear effects. Arguably the simplest dynamical system that emulates an excitable medium is a 3 state, range 1, threshold 1 cellular automaton known as the Greenberg Hastings model (GHM) (cf. [GH]). Over the past few years, in joint work with Robert Fisch and Janko Gravner, we have carried out a detailed experimental study of a three-parameter family of simple GHM-type rules. The parameters are the range ρ of interaction (assuming a box neighborhood N = {x Z 2 : x 1}), the threshold number θ of excited neighbors required for a cell to become excited, and the number κ of possible states (colors) per cell. Here state 0 is rested, 1 is excited, and 2,..., κ 1 are refractory. A rested cell becomes excited by contact whenever it finds at least θ excited cells within its range ρ (box) neighborhood. The refractory states advance automatically each time, finally cycling from κ 1 back to 0, so κ governs the recovery time. In symbols, ξ t (x) = 0 means the medium is rested (excitable) at x at time t; ξ t (x) = 1 means the medium is excited at x at time t; ξ t (x) = 2,..., κ 1: the medium is recovering (refractory) at x at time t; and the deterministic update rule is: 0 1 at x iff θ 1 s within x + ρn (the range ρ neighborhood of x); for k 1, k k + 1 (mod κ) deterministically. From appropriate simple initial conditions these rules generate periodic traveling waves, in much the same way that The Wave propagates across the crowd at a rock concert or sporting event. From random or disordered configurations the same rules often exhibit complex self-organization characterized by the emergence of large-scale structure. For suitable θ, nucleating spiral cores lead to a locally periodic final state in which every site eventually cycles with period κ, but sites slaved to distinct cores are typically out of phase. Figure 2 shows a representative case: ρ = 8, θ = 28, κ = 8, started from uniform product measure π over the available colors. We refer to π affectionately as primordial soup. The array here, and for all graphics in this article, is , with some cropping at the left and right edges. In the realization that produced Figure 2 most of the system quickly relaxed to the 0-state, but five spiral

7 SELF-ORGANIZATION OF RANDOM CELLULAR AUTOMATA 55 Fig. 2. Nucleation of spirals in a Greenberg Hastings model. cores, sometimes called ram s horns, nucleated from the soup. At the time t = 40 shown, the spirals are in the process of spreading over the entire lattice. Note how wave fronts from distinct centers annihilate when they collide. Evidently color computer graphics provide an effective way to visualize complex multitype interacting systems. Progress in understanding the phenomenology of excitable cellular automata would have been almost impossible without extensive use of efficient parallel devices such as the CAM-6 Cellular Automaton Machine [TM]. Our ability to interact with CAM-6 evolutions on the fly has been particularly illuminating. Clearly, computer simulations are most helpful for answering the question how does system X behave? before one tries to prove a theorem about X. As technology improves, though, it is increasingly apparent that visualization of complex system dynamics actually augments the traditional deductive process as well. Let us summarize the highlights of our recent and ongoing theoretical research on excitable cellular automata, especially as it relates to the prototypical dynamics of two-dimensional spiral formation shown in Figure 2. We discovered in [FGG1] that as one varies the parameters, GHM displays a remarkably complex phase portrait containing several cutoffs that divide the ergodic behavior of the infinite system into qualitatively distinct regimes. For instance, one regime is characterized by statistical noise, another by the nucleation of stable spiral pairs shown in Figure 2, a third by clustering of aligned parades of wave fragments

8 56 DAVID GRIFFEATH (macaroni), and a fourth by global relaxation. Closely related Cyclic Cellular Automaton (CCA) models (cf. [FGG0]), in which every color updates by threshold contact with its successor, exhibit even more exotic behavior. CCA dynamics are also described in [FGG1], a largely empirical paper filled with color graphics, experimental data, and a host of conjectures. However the complexity of excitable dynamics should not give the impression that rigorous mathematics is hopeless! Many GHM and CCA rules admit finite configurations ξ(λ) known as stable periodic objects (spo s): arrangements in which the color at each x sees at least θ sites of its successor color within (x + ρn ) Λ. A moment s thought reveals that such a structure ξ(λ) cycles deterministically no matter what the configuration off Λ. For example, the cores of each spiral pair in Figure 2 are spo s. Since such structures must exist somewhere in the infinite primordial soup by the monkey-at-the-typewriter principle, and since such period κ spiral cores serve as pacemakers for their disordered environment, we should expect a locally periodic limiting state whenever spo s exist. At least in the θ = 1 case, this is a theorem: spo s are simply loops of sites along which the colors appear cyclically, and the simple proof in [FGG0] works equally well for GHM and CCA. A promising discovery of [FGG1] is the emergence of curvature and limiting dynamics in excitable CA systems as the threshold θ and range ρ increase, with θ/ρ 2 λ, say. This threshold range scaling is particularly appealing from a mathematical point of view since the limiting Euclidean evolutions are surprisingly amenable to rigorous analysis. For instance, Durrett and Griffeath [DG] investigate the geometry of spiral cores in the threshold range limit. Contact updating in R 2 is formulated as follows. At any discrete time t, each site x in the plane inspects a Borel neighborhood x + N, where N is the unit ball with respect to some Euclidean norm, and asks whether the area painted with its successor color exceeds θ. Then the entire continuum updates in one truly massive parallel computation. If the state of the system at time t is a random tessellation of space into connected color-components with smooth boundaries, then the configuration at time t + 1 will also be such a tessellation. Moreover, the action of interfaces is described by integral transformations that can be studied analytically. Using this scaling, in the case of the unit l -box N, one can construct spiral cores that are spo s for λ < , and also argue heuristically but persuasively that spiral cores cannot exist for λ > 2 3. Thus there is a critical point λ c, known as bend and observed empirically to be about 0.653, below which GHM produces locally periodic patterns in the spirit of Figure 2, but above which the ergodic behavior is altogether different. See [DG] for further details, including a very concrete algorithm for the construction of huge spirals. Fisch, Gravner, and Griffeath [FGG2] study the asymptotic frequency of nucleation in GHM dynamics as the number of colors κ becomes large. Starting from primordial soup, and assuming that the excitation threshold θ is not too large, the box size needed for formation of a spiral core is shown to grow exponentially in κ. By exploiting connections with percolation theory, the exponential scaling rate is rigorously determined as 0.23 ±.06 in the nearest neighbor, threshold 1 case. By way of contrast, GHM rules obey power-law nucleation scaling when started from a suitable non-uniform product measure over the colors; this effect is driven by critical percolation. Along with the proofs, [FGG2] contains a nice picture of percolating

9 SELF-ORGANIZATION OF RANDOM CELLULAR AUTOMATA 57 spiral formation. Gravner and Griffeath [GG1] calculate the asymptotic shape of excitable CA nucleation droplets, on integer lattices and in Euclidean space. The limit shape L is identified as the polar transform of an explicitly computable width function. Even though the edges of droplets in Figure 2 appear smooth, L is an explicitly computable polygon with a very large number of sides in this case. In fact, by formulating an abstract version of the main limit theorem in [GG1], one can show that L is always a polygon for threshold growth CA rules on Z d. However subtle problems remain to be addressed for GHM dynamics that generate spreading rings. Such a ring is unstable if too thin, or if its curvature is too great at some location. Starting from a sufficiently large ring with uniformly small curvature, the wave should nevertheless be able to spread indefinitely. Delicate issues of boundary behavior make this a challenging problem. It is hoped that connections with curvature-driven partial differential equations will prove useful; a related connection is described in our third snapshot. Having studied spiral cores, nucleation density and droplet growth, it is natural to ask next What is the geometry of the final locally periodic state for typical GHM dynamics in the limit of rare nucleation (e.g., κ )? In a forthcoming project, Griffeath and Gravner [GG2] will conjecture for GHM, and prove for simpler excitable dynamics, that under suitable rescaling this limiting field is a Poisson Voronoi Tessellation (PVT) with respect to the norm that describes droplet growth. Recall that a PVT is a tessellation of Euclidean space such that the centers of the individual tiles constitute a Poisson field, and such that each tile comprises those locations which are closest to a given center with respect to a prescribed norm. Our methods for proving convergence to PVT naturally combine nucleation analysis, Poisson approximation, and shape theory. An example of a CA that we can handle rigorously is the Competing Growth Model on {0,..., κ 1}, where ξ t (x) = k means the medium has opinion (color) k at x at time t, (0 designating undecided voters), and the update rule is: 0 k at x iff θ k s within x + ρn, and k is the only such color. Otherwise there is no change at x; in particular, no k 1 ever changes. Figure 3, an even simpler Multitype Threshold Voter Model with box neighborhood, ρ = 2, θ = 7, and κ = 32, should convey the spirit of our results. (This variant has no background state 0.) If nucleation is rare, then the locations of nucleating centers will be approximately Poisson. Moreover, individual droplets are sufficiently separated that they nearly attain their limiting shape L before they interact. Finally, interaction is the simplest possible: a standoff between competing droplets wherever they meet. On the scale of mean distance between nucleating centers, with a suitable formulation of tile boundaries, and adopting the proper weak convergence framework, convergence to PVT should follow.

10 58 DAVID GRIFFEATH Fig. 3. Nucleation of random tiles in a Multitype Threshold Voter Model. We cannot discuss further details here. However one special case of our result is so simple that it is nearly transparent. With nearest neighbors and threshold 1, any non-zero color grows a diamond-shaped droplet that captures every site to which it is closest in the -norm. We invite the reader to argue for the following Theorem 1. Assume θ = 1. Start the nearest neighbor Competing Growth Model from a soup with density p/(κ 1) of each color k 1, and density 1 p of 0 s. Let C p,κ denote the set of sites in Z 2 that eventually have more than one color in their neighborhood. Then as p 0 and κ, p Cp,κ V(P), where V(P) is the Poisson Voronoi Tessellation for the -norm. 4. Euclidean Majority Vote: Curvature-Driven Clustering It is hard to imagine a simpler self-organizing scheme than majority vote. Citizens of two political persuasions, say Conservative and Labour, populate the lattice. From time to time individuals poll their neighborhood, succumb to peer-group pressure, and affiliate themselves with the local majority. Assuming a symmetric neighbor set

11 SELF-ORGANIZATION OF RANDOM CELLULAR AUTOMATA 59 Fig. 4. Self-organization of a Majority Vote rule after 3 updates. N, and counting one s own previous opinion in the tally, there is no chance of a tie so the algorithm is well-defined. In symbols, ξ t (x) = k {0, 1} means the system has opinion (color) k at x at time t, and the range ρ update rule is: Switch opinion at x iff > 1 2 of the voters in neighborhood x + ρn have the opposite opinion. (4) Figures 4 and 5 show a typical realization of the range 4 box Majority Vote CA, at times 3 and 10 respectively, started from symmetric product measure. Experiments such as this indicate clustering: from disordered noise the system appears to find a random tessellation within 2 or 3 updates, after which it self-organizes on length scales that grow over time. Real-time visualization reveals a surface tension effect. Minority components are eroded most rapidly along sections of the boundary where curvature is greatest. Small convex pockets of opposition are eliminated rapidly, but larger ones take much longer to disappear. In truth, the system in Figures 4 5 will fixate once the curvature of the tile boundaries is sufficiently small over the entire lattice. Since 41 of the 81 neighbors must disagree for a flip to occur, any tessellation with edges sufficiently close to flat is stable. For the range ρ version of majority vote

12 60 DAVID GRIFFEATH Fig. 5. The same Majority Vote CA after 10 updates. with stochastic dynamics, fixation can actually be proved by an energy argument ([DS, Dur2]). In order to sustain surface tension clustering indefinitely one must increase ρ without bound. Let us therefore consider the threshold range scaling limit known as Euclidean Majority Vote (EMV). Our update rule is the same as (4), but with x R 2, ρ = 1, and symmetric convex N R 2. For instance, isotropic dynamics are obtained by choosing N to be the Euclidean unit ball. Of course now the majority condition is phrased in terms of Euclidean area, an interpretation that certainly makes sense if we start from a random two-colored tessellation with smooth boundary. Moreover, one can check that the EMV update rule maps a suitable space of such tessellations into itself. It is a bit more challenging to formulate the continuum counterpart of primordial soup. Roughly speaking, Bernoulli product measure should become White Noise. Alternatively note that, on the lattice, {ξ 1 = 1} may be viewed as the positive part of a (correlated) random field. Passing to the continuum, confused sites of ξ 1 become the zero set of a (correlated) Gaussian field. It is plausible, then, and suggested by large-range lattice experiments, that ξ 2 should consist of countably many connected color components with continuous boundary. Only a flat edge is stable for EMV, so once the system nucleates a random tessellation, surface-tension clustering should continue indefinitely. On the basis of these heuristics, we offer the following bold

13 SELF-ORGANIZATION OF RANDOM CELLULAR AUTOMATA 61 Conjecture 1. Starting from White Noise, Euclidean Majority Vote nucleates components with continuous boundaries by time t = 2, and then clusters to arbitrarily large length scales as t. That is to say, for any bounded Λ R 2, as t, P (ξ t has one opinion on Λ) 1. (5) In current joint work with J. Gravner, we are attempting to make some headway on this conjecture. One key ingredient is a connection with Motion by Mean Curvature (MMC), a p.d.e. for which a rich and detailed theory has been developed over the past few years (cf. [ES]). Starting from tessellations with large length scale, one can show that EMV dynamics are well-approximated by Motion by Mean Curvature. More precisely, if γ is smooth and simple in R 2, [γ] its bounded component, then as n, for a suitable uniform sense of convergence, 1 n ξ n 2 t(n[γ]) [γ t ], (6) where γ t t = 1 6 φ2 (n)κ. (7) Here n is the unit normal, κ is the curvature, and φ(n) is the radius of N in the direction perpendicular to n. This last equation describes anisotropic MMC. The idea behind its derivation is rather straightforward. Half-spaces are invariant for EMV. The boundary of nγ has small curvature and so is well-approximated locally by a parabolic arc. A simple exercise in calculus shows that the amount nγ moves in direction n turns out to be proportional to κn 2. Another way to think about this approximation, by rescaling, is to fix γ and let the neighbor set n 1 N shrink to a point. In this way, the integral averaging of MVT reduces to the local operator for MMC. We note in passing that numerical analysts sometimes use generalizations of MVT as parallel schemes for simulation of equation (7). Using (6), we are able to prove a result that captures some of the ingredients in Conjecture 1. Here we merely outline the proof; details will appear elsewhere. Theorem 2. Starting from the symmetric Bernoulli 2-coloring of a sufficiently large honeycomb lattice, Euclidean Majority Vote clusters. Sketch of proof. Standard techniques in percolation theory (see [Gri]) imply that the connectivity of the 2-colored honeycomb is critical (site percolation on the hexagonal lattice has p c = 1 2 ). As a consequence, its connected color components form an infinite cascade : any component is surrounded by a circuit of the opposite color. A simple lemma shows that EMV (or MMC) dynamics preserve separation, i.e., there is an L < such that any component initially isolated from other components of the same color by distance L will remain so at all times (for MMC any L > 0 has this property). Isolation in the initial large honeycomb effectively precludes interaction between components (or contours) and, together with monotonicity, implies that every component eventually shrinks to at least as quickly as the smallest ball that covers it. Isotropic MMC is curve shortening; by approximation (6) the lengths of large EMV contour boundaries are controlled. Hence boundary length per unit area must tend to 0 as t, a property that implies (5).

14 62 DAVID GRIFFEATH Fig. 6. Self-organization of a Plurality Vote CA. We view the honeycomb lattice as being situated at a critical point toward which EMV nucleates. Very roughly, locations in the random tiling which are not wellseparated should disappear rapidly under iteration of the update rule. In this sense, EMV started from White Noise would appear to be self-organized critical. There are difficult obstacles, both conceptual and technical, to our understanding of the nucleation mechanism. But the phenomenology of surface tension clustering from disordered initial states is of broad interest (e.g., in the so-called spinodal decomposition of Stochastic Ising Model phase transitions), so even partial results seem worthwhile. Going way out on a limb, our heuristics even suggest the possibility of a limiting Euclidean random field statistically self-similar under MMC. A last remark in connection with this snapshot: suppose additional political parties enter the fray so that our conformist voter is confronted with a multitude of candidates. The natural Plurality Vote rule chooses the clear favorite over the neighborhood, but stays with the current opinion in case of a tie. How does this CA self-organize from κ color primordial soup? Figure 6 shows a simulation reminiscent of soap bubble patterns, with ρ = 6 (box neighborhood), κ = 15, at time t = 100. Again the interfaces emulate MMC, and again the system presumably clusters, although the nature of the clustering is quite different from the two-color case.

15 SELF-ORGANIZATION OF RANDOM CELLULAR AUTOMATA Larger than Life: Evolution of Complex Local Structure Our fourth and final snapshot is even more speculative. Together with graduate student Kellie Evans, we are studying a four-parameter family of cellular automata that generalize John Conway s celebrated Game of Life to the higher range, general threshold context. Recall that Conway s Life, probably the most famous of all CA rules, is a range 1 box, single species population model with remarkably complex dynamics. A birth can only occur at a cell with exactly 3 occupied neighbors, while survival requires either 2 or 3 occupied neighbors. See [BCG] for an entertaining account of both the recreational and theoretical study of Life. One should bear in mind that the early investigations of Conway and his cohort, as popularized by Martin Gardiner in Scientific American, predated the advent of desktop computer visualization. Cambridge veterans tell me that the first experiments were carried out on Go boards equipped with remarkably reliable cerebral processing units. Now that simulations of Life can be found on any respectable electronic bulletin board, and appear as the default screensaver on many SparcStations, it is easy to check that Conway s choice of the parameter values (3, 2 or 3) for birth and survival generate the most intriguing dynamics of any of the range 1 population rules of the same general form. We wondered whether his rule might be a clue to a critical phase point in the threshold range scaling limit. Thus, the Larger than Life (LTL) family of cellular automata have ξ t (x) {0, 1}; 1 means a creature lives at x at time t, 0 not. The update rule is: A birth at x iff the population on (x + ρn ) lies in [β, β + ]; a death at x unless the population on (x + ρn ) lies in [δ, δ + ]. Special cases of LTL have been considered in [Ruc] (ρ = 1) and [BBC] (ρ = 1, β ± = δ ± ). Extensive simulation of representative rules from the four-parameter family reveals a surprisingly rich phase space filled with many qualitatively distinct instances of nucleation and self-organization. The terrain is much more difficult to map out than that of GH/CCA because initial configurations far removed from primordial soup are often needed to sustain life. For now, let us simply offer a few illustrations. First, we looked at large-range CA rules near Conway s Life under threshold range scaling. Of course ρ = 1 is a very small parameter value, but it is not unreasonable to interpret Conway s rule as the LTL case [β, β + ] = [2.5, 3.5], [δ, δ + ] = [2.5, 4.5]. A natural scaling scheme is to identify rules with the same values of parameters/ ρn. Thus Conway s rule has approximately the values [.28,.39], [.28,.5] in the phase space. Figure 7 shows a still frame of the range 12 box LTL rule with (integer) parameters [β, β + ] = [170, 240], [δ, δ + ] = [170, 296]. We leave it to the reader to check that the position in our numerological phase space is quite close. Evidently our large-range rule generates complexity that is reminiscent of the original Life. In particular there are finite periodic structures, akin to Conway s blinkers, that move through a sequence of basic geometric shapes, and there are mobile bugs

16 64 DAVID GRIFFEATH Fig. 7. Larger than life: emergence of bugs (aka gliders). with an invariant shape, akin to Conway s celebrated gliders. We have discovered essentially the same phenomenology for rules up to range 15 in a small neighborhood of an apparent critical point. The resulting bugs seem to settle down to a limiting shape with a fat head, slender posterior, and a stomach, as shown in the Figure 7 inset. An entirely different self-organized evolution occurs for the range 15 LTL rule with parameters [β, β + ] = [170, 240], [δ, δ + ] = [170, 296], as illustrated in Figure 8. In this case most of the original soup dies out, but various small configurations are viable. Among these are rings of a characteristic diameter and band width that cannot spread on their own. However interactions between two or more such rings create web-like structures that nucleate to cover the entire space with a complex statistical equilibrium. We call this highly non-linear scenario nucleating pretzels. LTL dynamics can display many other exotic forms of self-organization. The phenomenology is so diverse and bewildering that we have decided to focus on some special cases. The exactly θ rule is particularly easy to state: there is a 1 at x next time iff there are exactly θ 1 s in the neighborhood of x this time, excluding x itself. Since updates require an exact population count over the neighbor set, this CA is rather different in spirit from our previous examples, and any viable patterns of occupied sites are necessarily more one-dimensional in spirit. By simply asking whether finite configurations of 1 s can survive and propagate we discover a series

17 SELF-ORGANIZATION OF RANDOM CELLULAR AUTOMATA 65 Fig. 8. Larger than life: complex nucleation of pretzels. of apparent critical phenomena: If 1 θ ρ, a suitable finite segment of occupied cells (vertically oriented, say) self-replicates and gives rise to a spreading fractal-like structure. This claim is actually a little theorem since, despite the non-linear rule, the dynamics mimic the genesis of a Sierpinski lattice. For θ ρ + 1, from the same initial seed, orderly propagation breaks down. Instead, for values of θ just above ρ, complex growth is reminiscent of a snowflake or Rorschach test. For θ just below 2ρ growth is apparently no longer possible, but viable periodic bugs emerge. We call these bugs skeeters since one of their characteristic shapes consists of a small solid head with two long trailing onedimensional legs. We have observed skeeters in rules up to range 40. Some of these bugs have very long periods; a few are even capable of giving birth to new skeeters that travel in the opposite direction. This last effect is reminiscent of procreation by glider guns in Conway s Life. No propagation appears possible for θ > 2ρ, although one can exhibit finite fixed structures for 3 θ (ρ + 1) 2 1. Global death occurs from any initial configuration once θ is large enough, e.g., for θ > 2ρ 2 + 3ρ. This little theorem is proved by comparison with monotone threshold growth.

18 66 DAVID GRIFFEATH In conclusion, let us mention some subtle open problems motivated by recent controversy surrounding Conway s Game of Life. In [BCC], a cover article of Scientific American, and elsewhere, it has been suggested that Life may be self-organized critical, a claim that includes power-law decay of the density of sites that are neither in the 0 state nor part of some periodic local configuration. Cellular Automaton Machine experiments of [BB], to the contrary, indicate relaxation at a small exponential rate. To a first approximation, Life may be viewed as an interaction between gliders/bugs mobile finite structures of fixed size and shape; and blinkers periodic immobile configurations. Interactions between any pair of such objects typically destabilizes both, leading to mutual annihilation. As a prototype, one may consider a system of 2 d billiards that move in one of the four directions N, S, E, W at each update, and fixed obstacles occupying one cell each, with annihilation upon any collision. If the initial density of billiards is p (p/4 for each type), and the initial density of obstacles is q, we may ask how the density of billiards tends to 0 over time, as a function of the parameters p and q. In the infinite system it is conjectured that the asymptotic rate is always exponential, but only after an initial transient period of apparent power-law decay that can be quite long if p exceeds q. In joint work with Maury Bramson, we will attempt to obtain rigorous results along these lines, and also to investigate the impact of this phenomenon on the behavior of corresponding finite-lattice systems. Perhaps such an analysis will help shed light on the above-mentioned controversy and indicate some important issues of scale in the approximation of infinite complex systems by the finite ones that are used for computer experimentation. However, as noted in [BB], one should not rule out the possibility that Life is actually supercritical! Namely, it is conceivable that Conway s game admits indestructible local configurations similar to the spo s of excitable cellular automata: exceedingly rare, perfectly synchronized constellations that send out impenetrable streams of colonists. Some day the offspring of such a monster might just show up on our doorstep and take over the world. References [BCC] Bak, P., Chen, K., and Creutz, M. (1989). Self-organized criticality in the Game of Life. Nature 342, [BB] Bennett, C. and Bourzutschky, M. (1991). Life not critical? Nature 350, 468. [BCG] Berlekamp, E., Conway, J., and Guy, R. (1982). Winning Ways for Your Mathematical Plays, Vol. 2. Academic Press, New York. [BBC] Bidaux, R., Boccara, N., and Chaté, H. (1989). Order of the transition versus space dimension in a family of cellular automata. The Physical Review A 39, [Dur1] Durrett, R. (1991). Probability: Theory and Examples. Wadsworth & Brooks/Cole, Pacific Grove, CA. [Dur2] Durrett, R. (1993). Ten lectures on particle systems. To appear as 1993 Saint-Flour Probability Summer School Lecture Notes, Springer-Verlag, New York. [DG] Durrett, R. and Griffeath, D. Asymptotic behavior of excitable cellular automata. Journal of Experimental Mathematics 3, to appear. [DS] Durrett, R. and Steif, J. (1993). Fixation results for threshold voter systems. Annals of Probability 21, [ES] Evans, L. C. and Spruck, J. (1993). Motion of level sets by mean curvature I. Journal of Differential Geometry, to appear.

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