Constant-length random substitutions and gibbs measures

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 substitutions rule, the existence of a unique process which remains invariant under the substitution, and which exhibits a 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 end up the paper by studying a class of substitutions whose invariant state is the unique Gibbs measure for a hierarchical two-body interaction."