Parsimonious Description of Generalized Gibbs Measures : Decimation of the 2d-Ising Model

TL;DR

This paper characterizes the decimation of the 2D Ising model within the generalized Gibbs framework, providing a detailed description of its almost Gibbsian properties and potential decay behavior.

Contribution

It offers a complete characterization of the decimated 2D Ising model as an almost Gibbsian measure with a quenched correlation decay potential, advancing the understanding of renormalized measures.

Findings

Decimated measures are almost Gibbsian with a well-defined potential.

The potential exhibits exponential decay beyond a configuration-dependent length.

The results integrate decimated measures into the framework of parsimonious random fields.

Abstract

In this paper, we detail and complete the existing characterizations of the decimation of the Ising model on $\Z^2$ in the generalized Gibbs context. We first recall a few features of the Dobrushin program of restoration of Gibbsianness and present the construction of global specifications consistent with the extremal decimated measures. We use them to consider these renormalized measures as almost Gibbsian measures and to precise its convex set of DLR measures. We also recall the weakly Gibbsian description and complete it using a potential that admits a quenched correlation decay, i.e. a well-defined configuration-dependent length beyond which this potential decays exponentially. We use these results to incorporate these decimated measures in the new framework of parsimonious random fields that has been recently developed to investigate probability aspects related to neurosciences.