The random cluster model on a general innnite graph with bounded degree wired at innnity is considered and a \ghost vertex" method is introduced in order to explicitly construct random clusterâ€¦ (More)

We consider the Ising model with external eld h and coupling constant J on an innnite connected graph G with uniformly bounded degree. We prove that if G is nonamenable, then the Ising model exhibitsâ€¦ (More)

Recently Wilson [12] introduced an important new technique for lower bounding the mixing time of a Markov chain. In this paper we extend Wilsonâ€™s technique to find lower bounds of the correct orderâ€¦ (More)

This paper is an up-to-date introduction to the problem of uniqueness versus non-uniqueness of infinite clusters for percolation on Z d and, more generally, on transitive graphs. For iid percolationâ€¦ (More)

It is shown that if G is any connected graph on n vertices, then the cover time for a random walk on the Cartesian product graph G k is at most of order dN(log N) 2 for k = 2 and at most of order dNâ€¦ (More)

The cover time of a finite connected graph is the expected number of steps needed for a simple random walk on the graph to visit all the vertices. It is known that the cover time on any n-vertex,â€¦ (More)

The strong Rayleigh property is a new and robust negative dependence property that implies negative association; in fact it implies conditional negative association closed under external fieldsâ€¦ (More)