Percolation and Minimal Spanning Forests in Infinite Graphs

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].. Institute of Mathematical Statistics is collaborating with JSTOR to digitize, preserve and extend access to The Annals of Probability. The structure of a spanning forest that generalizes the minimal spanning tree is considered for infinite graphs with a value f(b) attached to each bond b. Of particular interest are stationary random graphs; examples include a lattice with iid uniform values f(b) and the Voronoi or complete graph on the sites of a Poisson process, with f(b) the length of b. The corresponding percolation models are Bernoulli bond percolation and the "lily pad" model of continuum percolation, respectively. It is shown that under a mild "simultaneous uniqueness" hypothesis, with at most one exception, each tree in the forest has one topological end, that is, has no doubly infinite paths. If there is a tree in the forest, necessarily unique, with two topological ends, it must contain all sites of an infinite cluster at the critical point in the corresponding percolation model. Trees with zero, or three or more, topological ends are not possible. Applications to invasion percolation are given. If all trees are one-ended, there is a unique optimal (locally minimax for f) path to infinity from each site. 1. Introduction. For a finite set V c d a Euclidean minimal spanning tree (MST) of V is a tree with site (that is, vertex) set V and minimal total length of all bonds (that is, edges). More generally, given a finite graph with