Percolation and the hard-core lattice gas model

Abstract

Recently a uniqueness condition for Gibbs measures in terms of disagreement percolation (a type of dependent percolation involving two realizations) has been obtained. In general this condition is sufficient but not necessary for uniqueness. In the present paper we study the hard-core lattice gas model which we abbreviate as hard-core model. This model is not only relevant in Statistical Physics, but was recently rediscovered in Operations Research in the context of certain communication networks. First we show that the uniqueness result mentioned above implies that the critical activity for the hard-core model on a graph is at least P c (1 − P c) , where P c is the critical probability for site percolation on that graph. Then, for the hard-core model on bi-partite graphs, we study the probability that a given vertex v is occupied under the two extreme boundary conditions, and show that the difference can be written in terms of the probability of having a ‘path of disagreement’ from v to the boundary. This is the key to a proof that, for this case, the uniqueness condition mentioned above is also necessary, i.e. roughly speaking, phase transition is equivalent with disagreement percolation in the product space. Finally, we discuss the hard-core model on Z d with two different values of the activity, one for the even, and one for the odd vertices. It appears that the question whether this model has a unique Gibbs measure, can, in analogy with the standard ferromagnetic Ising model, be reduced to the question whether the third central moment of the surplus of odd occupied vertices for a certain class of finite boxes is negative.