Marginal dynamics of probabilistic cellular automata on trees

TL;DR

This paper characterizes the marginal dynamics of probabilistic cellular automata on trees using a local-field equation, revealing measure dependence from tree symmetries rather than mean field effects.

Contribution

It introduces a novel stochastic recursion for marginal evolution on trees, leveraging symmetries and Markov properties to analyze complex interacting processes.

Findings

Marginal dynamics can be described by an autonomous stochastic recursion.

The measure dependence stems from tree symmetries, not mean field interactions.

Established a second-order Markov random field property for general graphs.

Abstract

We study locally interacting processes in discrete time, often called probabilistic cellular automata, indexed by locally finite graphs. For infinite regular trees and certain generalized Galton-Watson trees, we show that the marginal evolution at a single vertex and its neighborhood can be characterized by an autonomous stochastic recursion referred to as the local-field equation. This evolution can be viewed as a nonlinear or measure-dependent chain, but the measure dependence arises from the symmetries of the underlying tree rather than from any mean field interactions. We discuss applications to simulation of marginal dynamics and approximations of empirical measures of interacting chains on several generic classes of large-scale finite graphs that are locally tree-like. In addition to the symmetries of the tree, a key role is played by a second-order Markov random field property, which we establish for general graphs along with some other novel Gibbs measure properties.