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- W286064771 abstract "In bond percolation on an in nite locally nite graph G = (V;E), each edge is randomly assigned value 0 (absent) or 1 (present) according to some probability measure on f0; 1g. One then studies connectivity properties of the random subgraph of G which arises by removing each edge with value 0. Maximal connected components of that subgraph are called clusters, and of particular interest is the possible existence of in nite clusters. Here we focus on the case where G is the regular tree Tn of order n 2. That is, Tn is the (unique) in nite connected graph that has no circuits and in which there are exactly n+ 1 edges emanating from each vertex. We write En and Vn for the edge and vertex sets of Tn. The most studied choice of probability measure on f0; 1g is i.i.d. measure. When G = Tn, this reduces to the study of Galton{Watson branching processes with binomial o spring distribution. Here we consider the more general class of automorphism invariant probability measures on f0; 1gn, i.e. measures that are invariant under graph automorphisms of Tn; a graph automorphism for Tn is a bijection : Vn ! Vn that preserves adjacency, together with the induced mapping 0 : En ! En. There are several interesting examples (besides i.i.d. measure) of such probability measures, including the random-cluster model and uniform spanning forests; see e.g. Haggstrom (1996, 1997, 1998). In Haggstrom (1997), we showed that any automorphism invariant probability measure on f0; 1gn whose marginal probability that an edge is present is at least 2=(n + 1), produces in nite clusters with positive probability (this bound was also shown to be sharp). The proof involved a mass-transport argument, which was extended and exploited with great success by Benjamini, Lyons, Peres and Schramm (1998a). The mass-transport method is discussed in Section 2. Several results concerning the number and topological structure of in nite clusters were also given in Haggstrom (1997). For space reasons, some of the proofs were omitted in the nal version of that paper. In Section 3 we shall recall these results and give new proofs, which are short and simple, based on the mass-transport method." @default.
- W286064771 created "2016-06-24" @default.
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- W286064771 date "1999-01-01" @default.
- W286064771 modified "2023-09-27" @default.
- W286064771 title "Invariant percolation on trees and the mass-transport method" @default.
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