  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.
