We show that for large dimension d, the hard-core model on the integer lattice Z^d with all activities equal to lambda exhibits a phase transition when

More specifically:

We define the distance between any two points in the lattice to be the L^1 distance and say a subset I of the lattice is independent if no two points of I are adjacent (at distance 1).

Let B = [-N,N]^d be a large box in the lattice. For lambda >0, we consider the probability distribution on the independent subsets of B where each such I has probability proportional to lambda^|I|. Let p_e (resp. p_o) be the probability that I contains the origin given that all points on the boundary of B lying at an even (resp. odd) distance from the origin are in I.

We show that for N large and for lambda in the range above, these quantities are very different; namely, p_e is very close to lambda/(1 + lambda), whereas p_o = o(1/d).

This is joint work with Jeff Kahn.