# 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.

## Key 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.