Particle Systems arising from an anti-ferromagnetic Ising model

TL;DR

This paper investigates a low-temperature anisotropic anti-ferromagnetic 2D Ising model via a dimer model, revealing explicit phase diagrams and connections to particle systems, including the noisy voter model and Pfaffian point processes.

Contribution

It introduces a novel bijection between the Ising model and a one-dimensional particle system, and explicitly characterizes phase diagrams and limiting processes at critical and independent points.

Findings

Explicit phase diagrams as functions of temperature and anisotropy.

At independence, the particle system matches the two-colored noisy voter model.

At criticality, the particle distribution forms a Pfaffian point process with Bessel kernel.

Abstract

We study a low temperature anisotropic anti-ferromagnetic 2D Ising model through the guise of a certain dimer model. This model has a bijection with a one-dimensional particle system equipped with creation and annihilation. In the thermodynamic limit, we determine the explicit phase diagrams as functions of temperature and anisotropy. Two values of the anisotropy are of particular interest - the 'critical' value and the 'independent' value. At independence, the particle system has the same distribution as the two colored noisy voter model. Its limiting measure under a natural scaling window is the Continuum Noisy Voter Model. At criticality, the distribution of particles on a given horizontal line, is a Pfaffian point process whose kernel in the scaling window can be written explicitly in terms of Bessel functions.