Hitting times asymptotics for hard-core interactions on grids

TL;DR

This paper analyzes the asymptotic behavior of the first hitting time between maximum-occupancy configurations in the hard-core model on grid graphs, revealing how it depends on grid size and boundary conditions in the low-temperature regime.

Contribution

It introduces a novel combinatorial method to determine the order-of-magnitude of hitting times and extends existing frameworks to more general initial states and target sets.

Findings

Hitting times depend on grid size and boundary conditions.

Hitting times are asymptotically exponential in the low-temperature limit.

The paper determines the mixing time in the low-temperature regime.

Abstract

We consider the hard-core model with Metropolis transition probabilities on finite grid graphs and investigate the asymptotic behavior of the first hitting time between its two maximum-occupancy configurations in the low-temperature regime. In particular, we show how the order-of-magnitude of this first hitting time depends on the grid sizes and on the boundary conditions by means of a novel combinatorial method. Our analysis also proves the asymptotic exponentiality of the scaled hitting time and yields the mixing time of the process in the low-temperature limit as side-result. In order to derive these results, we extended the model-independent framework in [27] for first hitting times to allow for a more general initial state and target subset.