# Lack of phase transitions in staggered magnetic systems. A comparison of uniqueness criteria

## TL;DR

This paper investigates the conditions under which a ferromagnetic Ising model with a staggered magnetic field exhibits a unique Gibbs measure, using three different criteria to establish bounds and compare their effectiveness.

## Contribution

It provides the first rigorous comparison of three distinct criteria for the uniqueness of Gibbs measures in a staggered magnetic Ising model.

## Key findings

- Dobrushin criterion yields the broadest uniqueness region.
- Disagreement percolation provides tighter bounds in certain parameter ranges.
- Dobrushin-Shlosman criteria offer a different perspective on phase transition boundaries.

## Abstract

We study a ferromagnetic Ising model with a staggered cell-board magnetic field previously proposed for image processing [Maruani et al., Markov Processes Relat. Fields 1 (1995) \cite{MPS}]. We complement previous results on the existence of phase transitions at low temperature [Gonz\'alez-Navarrete et al., J. Stat. Phys. 162 (2016)] by determining bounds to the region of uniqueness of Gibbs measures. We establish sufficient rigorous uniqueness conditions derived from three different criteria: (1) Dobrushin criterion [Dobrushin, Theory Probab. Appl. 13 (1968)], (2) Disagreement percolation [van den Berg and Maes, Ann. Probab. 22 (1994)] and (3) Dobrushin-Shlosman criteria [Dobrushin and Shlosman, in Statistical Physics and Dynamical Systems. Rigorous Results. (1985)]. These conditions are subsequently solved numerically and the resulting uniqueness regions compared.