Odd cutsets and the hard-core model on ℤ^d

We consider the hard-core lattice gas model on ℤ^d and investigate its phase structure in high dimensions. We prove that when the intensity parameter exceeds Cd^-1/3(log d)², the model exhibits multiple hard-core measures, thus improving the previous bound of Cd^-1/4(log d)^3/4 given by Galvin and Kahn. At the heart of our approach lies the study of a certain class of edge cutsets in ℤ^d, the so-called odd cutsets, that appear naturally as the boundary between different phases in the hard-core model. We provide a refined combinatorial analysis of the structure of these cutsets yielding a quantitative form of concentration for their possible shapes as the dimension d tends to infinity. This analysis relies upon and improves previous results obtained by the first author.