Graded cluster expansion for lattice systems

TL;DR

This paper develops a unified framework using graded cluster expansion to analyze lattice systems, addressing weak Gibbs measures and mixing in disordered phases under certain mixing conditions.

Contribution

It introduces a general theory that unifies the treatment of weak Gibbs properties and mixing in disordered lattice systems through a convergent multi-scale cluster expansion.

Findings

Unified approach for Gibbs measures and mixing properties

Construction of convergent multi-scale cluster expansion

Applicable to systems with sparse large regions

Abstract

In this paper we develop a general theory which provides a unified treatment of two apparently different problems. The weak Gibbs property of measures arising from the application of Renormalization Group maps and the mixing properties of disordered lattice systems in the Griffiths' phase. We suppose that the system satisfies a mixing condition in a subset of the lattice whose complement is sparse enough namely, large regions are widely separated. We then show how it is possible to construct a convergent multi-scale cluster expansion.