Constant-length random substitutions and Gibbs measures

TL;DR

This paper investigates processes generated by constant-length random substitutions, proving the existence of a unique invariant process that is a Gibbs measure with polynomial decay of correlations, and explores specific classes with hierarchical Gibbs states.

Contribution

It establishes the existence and uniqueness of invariant processes for random substitutions and characterizes these as Gibbs measures in the constant-length case.

Findings

Unique invariant process exists under mild conditions

Invariant state is a Gibbs measure for constant-length substitutions

Identifies classes with hierarchical Gibbs states

Abstract

This work is devoted to the study of processes generated by random substitutions over a finite alphabet. We prove, under mild conditions on the substitution's rule, the existence of a unique process which remains invariant under the substitution, and exhibiting polynomial decay of correlations. For constant-length substitutions, we go further by proving that the invariant state is precisely a Gibbs measure which can be obtained as the projective limit of its natural Markovian approximations. We close the paper with a class of substitutions whose invariant state is the unique Gibbs measure for a hierarchical two-body interaction.