MEAN FIELD MODELS OF PROBABILISTIC CELLULAR AUTOMATA

Probabilistic cellular automata (PCA) form a very large and general class of stochastic processes. These automata exhibit a wide range of complicated behavior and are of interest in a number of fields of study, including mathematical physics, percolation theory, computer science, and neurobiology. The models are generally considered on the lattice Z (usually in two dimensions). Very little has been proved about these models, even in simple cases, so it is common to compare the models to mean field models. It is normally assumed that mean field models are essentially trivial. However, we show here that even the mean field models can exhibit surprising behavior. We also consider some of the quantitative differences between the mean field and lattice models. 1. Probabilistic Cellular Automata Let A be a finite set of states and let Γ be a finite subset of Z containing the origin 0 = (0, . . . , 0). Fix a function p : A × A → [0, 1] that assigns for each configuration φ : Γ → A and each a ∈ A a probability pφ,a with ∑ a∈A pφ,a = 1 for all φ. We define a sequence of configurations Φt : Z d → A by setting Φt+1(x) = a independently for each x ∈ Z with probability pφ,a where φ(y) = Φt(x+ y) is the restriction to Γ of (a translate of) Φt. We start the process with Φ0 given by some specified distribution on A d . We call such a process a probabilistic cellular automaton or PCA on Z. Usually we define Γ to be the closed neighborhood of 0, consisting of all points unit distance from 0 together with 0 itself, or more generally, all points within some fixed distance of 0. Similarly we can define probabilistic cellular automaton on the d-dimensional torus Zn1 × · · · × Znd, or on the corresponding finite grid (with suitable boundary conditions imposed). These models have also been referred to as contact processes and have been studied in some simple cases on infinite graphs [24, 25]. Probabilistic cellular automata generalize deterministic cellular automata such as Conway’s game of life. Given the complexity of deterministic automata it should not come as a surprise that random automata display extremely complex behavior which is extremely difficult to analyze rigorously. In the models described, we have updated all the sites in Z simultaneously at each step. One can ask what happens if the sites are updated asynchronously? Equivalently, run the process in continuous time t and change each site according to independent exponential random variables whose rates rφ,a (defined for a = φ(0)) depend on the Date: August 25, 2004. 1 2 P. BALISTER, B. BOLLOBÁS, AND R. KOZMA neighborhood of the site. For a finite grid the asynchronous model can be obtained as a limit of synchronous models as follows. Given transition rates rφ,a for φ ∈ A, define for sufficiently small > 0 the random automaton with probabilities pφ,a = { rφ,a if φ(0) = a 1− ∑b =a rφ,b if φ(0) = a (1.1) Then as → 0 this automaton approximates the asynchronous model with rates rφ,a since the chances of neighbors being updated at the same time tends to zero. For the infinite lattice Z one expects the same result to hold — the asynchronous model being approximated increasingly well by the synchronous model as → 0 (with suitable rescaling of t). As a result, asynchronous models can be considered as a special case of synchronous models. However, the synchronous models display a richer variety of behavior, such as small period oscillations, that are not possible in asynchronous models. Also, it is not clear that, for example, critical exponents at phase transitions for asynchronous models necessarily match those of the approximating synchronous models. There are however reasons for believing that the asynchronous models are easier to analyze — for example, the Kinetic Ising model can be more easily described by an asynchronous model. 2. The Mean Field Approximation From now on we shall restrict our attention to 2-state processes with A = {−,+} (or sometimes {0, 1}). In this case we write pφ for pφ,+ and so pφ,− = 1− pφ. We start by considering p of a certain special form. Assume pφ depends only on the cardinality of set of neighbors that are in state +, φ−1(+) = {x ∈ Γ : φ(x) = +}, and on the state of the site itself, φ(0). We shall call these models semi-totalistic. We write pr (resp. p − r ) in place of pφ when |φ−1(+)| = r and φ(0) = + (resp. φ(0) = −). Semi-totalistic models are substantially more restrictive than the general case, but they still have complex behavior, sometimes including spontaneous symmetry breaking (see below). We shall call the model totalistic if pr = p + r = pr for all r. In this case, the site itself is treated on the same basis as its neighbors. By interchanging + and − throughout we obtain a new model with pφ = 1 − p−φ (p′r = 1 − p|Γ|−r for semi-totalistic models). We call the model symmetric if it is the same under interchange of + and −, so pφ = 1 − p−φ. If we complement every other Φt, in a symmetric model we obtain a new symmetric model with pS replaced by pS = 1− pS. We call this new model the dual model of the symmetric model (pS). Symmetric models with approximate period two oscillations can often be more easily studied by looking at the dual model. On a finite grid or torus, we can compare probabilistic automata with the corresponding mean field models. In the mean field model instead of taking |Γ|−1 specified neighbors, we take |Γ| − 1 elements of the grid at random (with replacement). Since there is no ordering of these neighbors, the transition probabilities depend only on the number of + states in this neighborhood and the state of the site itself, so may MEAN FIELD MODELS OF PROBABILISTIC CELLULAR AUTOMATA 3 be given by probabilities pr as in the semi-totalistic case. Because of this, we shall only consider semi-totalistic models from now on. However, we shall not in general assume our models are totalistic, since even for a totalistic asynchronous model, the synchronous approximations described above and only semi-totalistic. It is clear that the mean field model does not depend on the topology of the grid, and the only information of relevance in Φt is given by the cardinality of +-states, Φ−1(+) = {x : Φt(x) = +}. We define ρt to be |Φ−1(+)|/N where N is the size of grid. Thus ρt ∈ [0, 1] gives the density of + sites in Φt. Write B(k, p) for a binomial random variable giving the sum of k independent Bernoulli random variables, each of which is 1 with probability p. Theorem 2.1. If there are N points in the grid then ρt is given by Nρt+1 = B(Nρt, f + m(ρt)) +B(N(1− ρt), f− m(ρt)), (2.1) where the functions f± m are defined by f m(x) = ∑ r (|Γ|−1 r−1 ) pr x r−1(1− x)|Γ|−r, (2.2)