Renormalization Group Transformations under strong mixing conditions: gibbsianess and convergence of renormalized interactions

TL;DR

This paper investigates the behavior of renormalization-group transformations on Gibbs measures for lattice gases under strong mixing conditions, proving Gibbsianess and convergence to a Gaussian fixed point, including the 2D Ising model above critical temperature.

Contribution

It establishes Gibbsianess and convergence of renormalized interactions for a broad class of lattice gases under strong mixing, extending to the 2D Ising model at high temperatures.

Findings

Proves Gibbsianess of renormalized measures

Shows convergence to Gaussian fixed point

Applicable to 2D Ising model above critical temperature

Abstract

In this paper we study a renormalization-group map: the block averaging transformation applied to Gibbs measures relative to a class of finite range lattice gases, when suitable strong mixing conditions are satisfied. Using block decimation procedure, cluster expansion (like in [HK]) and detailed comparison between statistical ensembles, we are able to prove Gibbsianess and convergence to a trivial (i.e. Gaussian and product) fixed point. Our results apply to 2D standard Ising model at any temperature above the critical one and arbitrary magnetic field.