Finitary coding and Gaussian concentration for random fields

TL;DR

This paper establishes conditions under which Gaussian concentration inequalities hold for random fields derived from i.i.d. processes via finitary codings, linking concentration properties to the coding structure and applying to various lattice models.

Contribution

It proves that Gaussian concentration is preserved under finitary codings with finite second moments and characterizes this property for classical lattice models and Markov chains.

Findings

Gaussian concentration preserved under finitary codings with finite second moment

Necessary and sufficient conditions for Gaussian concentration in lattice models

Characterization of processes with unbounded memory in terms of ergodicity and coding properties

Abstract

We study Gaussian concentration inequalities for random fields obtained as finitary codings of i.i.d.\ fields, linking concentration properties to coding structure. A finitary coding represents a dependent field as a shift-equivariant image of an i.i.d.\ process, where each output depends on a finite but configuration-dependent portion of the input. Gaussian concentration corresponds to uniform sub-Gaussian bounds for local observables. Our main abstract result shows that Gaussian concentration is preserved under finitary codings with finite second moment of the coding volume. The proof relies on a refinement of the bounded-differences inequality, due to Talagrand and Marton, handling configuration-dependent influences. Under an additional structural assumption, satisfied in particular by coupling-from-the-past codings, a finite first moment suffices. These moment conditions are sharp. We apply these results to Gibbs measures, Markov random fields on $\mathbb Z^d$, and a broad class of one-dimensional processes. Using recent constructions of finitary codings, notably by Spinka and collaborators, we obtain sharp necessary and sufficient conditions for Gaussian concentration in classical lattice models, including the Ising, Potts, and random-cluster models: it holds if and only if the model lies in the full uniqueness regime, extending previous results beyond strict subregimes. In one dimension, we treat processes with possibly unbounded memory. For countable-state Markov chains, we obtain equivalent characterizations in terms of geometric ergodicity, exponential return-time tails, and finitary i.i.d.\ codings with exponential tails.