|
Abstract : |
Abstract. We survey the field of uniform spanning forest measures on infinite graphs, which are weak limits of uniform spanning tree measures from finite subgraphs. These limits can be taken with free or wired boundary conditions. Among other results, Pemantle (1991) proved that in Z d, the free and wired spanning forests coincide and that they give a single tree iff d 4. The theory has developed considerably since then and found further connections to random walks, potential theory, harmonic Dirichlet functions, invariant percolation and amenability. A crucial new tool is an algorithm invented by Wilson (1996) to generate random spanning trees. Uniform spanning forests also yield insights into loop-erased walks and harmonic measure from infinity. x1. Introduction. We begin with a brief history of this fertile and fascinating field. In subsequent sections, we will more carefully define and develop most of what we recount here. More details for much of the material surveyed here can be found in Benjamini, Lyons, Peres,, |
