Renormalization Group in the uniqueness region: weak Gibbsianity and convergence

TL;DR

This paper investigates the behavior of lattice gas models below the critical temperature, demonstrating weak Gibbsianity and convergence of the renormalized potential using a scale-adapted expansion near the coexistence region.

Contribution

It introduces a novel approach to analyze the renormalization group in the uniqueness region, accounting for diverging correlation lengths and Griffiths' singularity effects.

Findings

Proves weak Gibbsianity of the renormalized measure.

Shows convergence of the renormalized potential in a weak sense.

Develops a scale-adapted cluster expansion near the coexistence line.

Abstract

We analyze the block averaging transformation applied to lattice gas models with short range interaction in the uniqueness region below the critical temperature. We prove weak Gibbsianity of the renormalized measure and convergence of the renormalized potential in a weak sense. Since we are arbitrarily close to the coexistence region we have a diverging characteristic length of the system: the correlation length or the critical length for metastability, or both. Thus, to perturbatively treat the problem we have to use a scale-adapted expansion. Moreover, such a model below the critical temperature resembles a disordered system in presence of Griffiths' singularity. Then the cluster expansion that we use must be graded with its minimal scale length diverging when the coexistence line is approached.