Markov Approximations of Gibbs measures for long-range potentials on 1D lattices

TL;DR

This paper investigates the approximation of Gibbs measures for long-range interactions on 1D lattices, establishing convergence results and mixing rate bounds under certain regularity conditions.

Contribution

It introduces a new regularity condition for long-range potentials that enables explicit bounds on mixing rates and convergence of entropy for Markov approximations.

Findings

Markov measures converge to the equilibrium state.

Explicit stretched exponential bounds on mixing rates.

Entropy of Markov measures converges to that of the equilibrium state.

Abstract

We study one-dimensional lattice systems with pair-wise interactions of infinite range. We show projective convergence of Markov measures to the unique equilibrium state. For this purpose we impose a slightly stronger condition than summability of variations on the regularity of the interaction. With our condition we are able to explicitly obtain stretched exponential bounds for the rate of mixing of the equilibrium state. Finally we show convergence for the entropy of the Markov measures to that of the equilibrium state via the convergence of their topological pressure.