Criticality of measures on 2-d Ising configurations: from square to hexagonal graphs

TL;DR

This paper investigates a family of non-Gibbsian measures on 2D Ising configurations, exploring how an inertial parameter affects critical behavior and correlations, linking square and hexagonal lattice models through the Random Cluster model.

Contribution

It introduces a new family of measures depending on an inertial parameter, connecting square and hexagonal lattice Ising models and analyzing their critical properties.

Findings

The measures are related to Gibbs measures on hexagonal lattices.

The inertial parameter q influences the system's geometry and critical behavior.

Correlation decay properties are characterized via the Random Cluster model.

Abstract

On the space of Ising configurations on the 2-d square lattice, we consider a family of non Gibbsian measures introduced by using a pair Hamiltonian, depending on an additional inertial parameter $q$. These measures are related to the usual Gibbs measure on $\Z^2$ and turn out to be the marginal of the Gibbs measure of a suitable Ising model on the hexagonal lattice. The inertial parameter $q$ tunes the geometry of the system. The critical behaviour and the decay of correlation functions of these measures are studied thanks to relation with the Random Cluster model.