  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.
