Ergodicity of probabilistic cellular automata: A constructive criterion

We give a sequence of criteria (of increasing complexity) for the exponential ergodicity of discrete time interacting particle systems. Each criterion involves estimating the dependence on initial conditions of the process on finite space-time volumes. It generalizes and improves the existing single site condition and is the analog of the Dobrushin-ShlosmanCv condition in equilibrium statistical mechanics. Our “dynamic” criterion may also be used to prove the uniqueness of Gibbs state in situations where theCv condition fails. As a converse we prove that if there is a certain form of convergence to the stationary measure faster thann−d, wheren is the time andd is the dimension of the lattice, then our condition holds for some space-time volumes and hence the convergence must be exponentially fast.

[1]  R. Dobrushin,et al.  Completely analytical interactions: Constructive description , 1987 .

[2]  William Feller,et al.  An Introduction to Probability Theory and Its Applications , 1967 .

[3]  P. L. Dobruschin.  The Description of a Random Field by Means of Conditional Probabilities and Conditions of Its Regularity , 1968 .

[4]  J. Lamperti.  Stochastic processes : a survey of the mathematical theory , 1979 .

[5]  T. Liggett.  Interacting Particle Systems , 1985 .

[6]  R. Dobrushin,et al.  Constructive Criterion for the Uniqueness of Gibbs Field , 1985 .

[7]  Roelof Kuik,et al.  From PCA's to equilibrium systems and back , 1989 .

[8]  J. Lebowitz,et al.  Statistical mechanics of probabilistic cellular automata , 1990 .

[9]  Lawrence Gray,et al.  The Behavior of Processes with Statistical Mechanical Properties , 1987 .

[10]  Unusual analytic properties of some lattice models: Complement of Lee-Yang theory , 1986 .

[11]  R. Holley,et al.  Rapid Convergence to Equilibrium of Stochastic Ising Models in the Dobrushin Shlosman Regime , 1987 .

[12]  Daniel F. Styer,et al.  Vasserstein distances in two-state systems , 1989 .

[13]  Hans-Otto Georgii,et al.  Gibbs Measures and Phase Transitions , 1988 .

[14]  S. Shlosman.  Uniqueness and half-space nonuniqueness of gibbs states in Czech models , 1986 .

[15]  F. Papangelou.  GIBBS MEASURES AND PHASE TRANSITIONS (de Gruyter Studies in Mathematics 9) , 1990 .

[16]  Daniel W. Stroock,et al.  Uniform andL2 convergence in one dimensional stochastic Ising models , 1989 .

[17]  Jeffrey E. Steif,et al.  -Convergence to equilibrium and space—time bernoullicity for spin systems in the M < ε case , 1991, Ergodic Theory and Dynamical Systems.