Critical configurations of the hard-core model on square grid graphs

TL;DR

This paper analyzes the transition mechanisms between two maximum-occupancy configurations in the hard-core model on square grid graphs at low temperatures, revealing complex critical configurations through geometric and isoperimetric analysis.

Contribution

It introduces a comprehensive geometric characterization of critical configurations and develops a novel isoperimetric inequality for hard-core configurations.

Findings

Characterization of critical configurations as asymptotically visited states

Development of a new isoperimetric inequality for fixed-particle configurations

Identification of shape and size as key factors in saddle configurations

Abstract

We consider the hard-core model on a finite square grid graph with stochastic Glauber dynamics parametrized by the inverse temperature $\beta$. We investigate how the transition between its two maximum-occupancy configurations takes place in the low-temperature regime $\beta\to\infty$ in the case of periodic boundary conditions. The hard-core constraints and the grid symmetry make the structure of the critical configurations, also known as essential saddles, for this transition very rich and complex. We provide a comprehensive geometrical characterization of the set of critical configurations that are asymptotically visited with probability one. In particular, we develop a novel isoperimetric inequality for hard-core configurations with a fixed number of particles and we show how not only their size but also their shape determines the characterization of the saddles.