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.
Highlights
1.1 Hard-Core Lattice Gas ModelIn this paper we consider a stochastic model where particles in a finite volume dynamically interact subject to hard-core constraints and study the first hitting times between admissible configurations of this model
1.4 Results for Rectangular Grid Graphs. We apply these model-independent results to the hard-core model on rectangular grid graphs to understand the asymptotic behavior of the hitting time τoe, where e and o are the two configurations with maximum occupancy, where the particles are arranged in a checkerboard fashion on even and odd sites
Using a novel powerful combinatorial method, we identify the minimum energy barrier between e and o and prove absence of deep cycles for this model, which allows us to decouple the asymptotics for the hitting time τoe and the study of the critical configurations
Summary
In this paper we consider a stochastic model where particles in a finite volume dynamically interact subject to hard-core constraints and study the first hitting times between admissible configurations of this model. This model was introduced in the chemistry and physics literature under the name “hard-core lattice gas model” to describe the behavior of a gas whose particles have non-negligible radii and cannot overlap [25,41]. We say that a particle configuration on is admissible if it does not violate the hard-core constraints, i.e., if it corresponds to an independent set of the graph.
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