Consider the random-cluster model on the integer lattice with parameters p and q. As p, q --> 0 in such a way that q/p --> 0, the random-cluster measures converge weakly to the uniform spanning tree measure of Pemantle (1991).

We consider the random coloring of the vertices of a graph G, that arises by first performing i.i.d. bond percolation with parameter p on G, and then assigning a random color, chosen according to some prescribed probability distribution on the finite set {0,...,r-1}, to each of the connected...

We study the two-type Richardson model on , d[greater-or-equal, slanted]2, in the asymmetric case where the two particle types have different infection rates. Starting with a single particle of each type, and fixing the infection rate for one of the types, we show that mutual unbounded growth...

We consider random walk with a nonzero bias to the right, on the infinite cluster in the following percolation model: take i.i.d. bond percolation with retention parameter p on the so-called infinite ladder, and condition on the event of having a bi-infinite path from -[infinity] to...