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  for first hitting times to allow for a more general initial state and target subset. Keywords: hard-core model; hitting times; Metropolis Markov chains; finite grid graphs; mixing times; low temperature.
Nardi, F. R., Zocca, A., & Borst, S. C. (2016). Hitting time asymptotics for hard-core interactions on grids. Journal of Statistical Physics, 162(2), 522-576. https://doi.org/10.1007/s10955-015-1391-x