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

A hard-core configuration on the 14x14 grid graph (left) and its equivalent representation only with odd clusters (right).


We consider the hard-core model on a finite square grid graph with stochastic Glauber dynamics parametrized by the inverse temperature β. 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.

S. Baldassarri, V. Jacquier, A. Zocca. (2023) Critical configurations of the hard-core model on square grid graphs. Submitted to Combinatorics, Probability and Computing.
Alessandro Zocca
Alessandro Zocca
Tenured Assistant Professor