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Marginal dynamics of probabilistic cellular automata on trees

Daniel Lacker, Kavita Ramanan, Ruoyu Wu

TL;DR

The paper introduces local-field equations that autonomously characterize the marginal distribution of a single vertex and its neighborhood for probabilistic cellular automata on infinite regular trees and (generalized) Galton–Watson trees. By exploiting tree symmetries and a second-order Markov random field property, the authors derive recursive constructions that couple a finite neighborhood to a history-dependent kernel, enabling efficient simulation and accurate approximation of macroscopic observables on locally tree-like graphs. They establish Gibbs-type uniqueness and a consistency principle, and extend the framework to unimodular GW trees, providing a unified approach to finite-graph limits via local convergence. The results yield practical algorithms for marginals and offer insights into the structure of interacting particle systems on sparse networks with broad applicability to networked dynamics and large deviations.

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.

Marginal dynamics of probabilistic cellular automata on trees

TL;DR

The paper introduces local-field equations that autonomously characterize the marginal distribution of a single vertex and its neighborhood for probabilistic cellular automata on infinite regular trees and (generalized) Galton–Watson trees. By exploiting tree symmetries and a second-order Markov random field property, the authors derive recursive constructions that couple a finite neighborhood to a history-dependent kernel, enabling efficient simulation and accurate approximation of macroscopic observables on locally tree-like graphs. They establish Gibbs-type uniqueness and a consistency principle, and extend the framework to unimodular GW trees, providing a unified approach to finite-graph limits via local convergence. The results yield practical algorithms for marginals and offer insights into the structure of interacting particle systems on sparse networks with broad applicability to networked dynamics and large deviations.

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.
Paper Structure (32 sections, 16 theorems, 154 equations)

This paper contains 32 sections, 16 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Informal derivation of the local-field equations on $\mathbb{Z}$
  3. Additional features of the local-field equations on trees
  4. Conditional independence and Gibbs measures
  5. Background and motivation
  6. Organization of the paper
  7. Statements of main results
  8. Common notation and terminology
  9. Space of unordered terminating sequences
  10. The local-field equation on a regular tree
  11. Local-field equations on Galton-Watson Trees
  12. Ulam-Harris-Neveu labeling of trees
  13. Dynamics and Results on Galton-Watson Trees
  14. The local-field equation on a unimodular Galton-Watson tree
  15. Approximation of dynamics on finite graphs
  16. ...and 17 more sections

Key Result

Theorem 2.3

Suppose Condition cond:regular_tree holds, and let $X$ be as in eq:regular_tree. Let ${\bar{X}}$ be as in Construction constr:regulartree_local. Then $(X_v)_{v =0}^{\kappa} \stackrel{d}{=} ({\bar{X}}_v)_{v=0}^{\kappa}$.

Theorems & Definitions (43)

  • Theorem 2.3: Local-field equations characterize marginals on regular trees
  • Remark 2.4
  • Remark 2.5
  • Remark 2.7
  • Example 2.8
  • Remark 2.10
  • Theorem 2.11: Local-field equations characterize marginals on GW trees
  • Remark 2.14
  • Theorem 2.15: Reduced local-field equations for UGW trees
  • Theorem 2.16: Characterization of limits of global empirical measures
  • ...and 33 more