Ends in Uniform Spanning Forests

By Russell Lyons, Benjamin J. Morris, Oded Schramm. 2007

51 citations recorded by the source corpus.

It has hitherto been known that in a transitive unimodular graph, each tree in the wired spanning forest has only one end a.s. We dispense with the assumptions of transitivity and unimodularity, replacing them with a much broader condition on the isoperimetric profile that requires just slightly more than uniform transience.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/00fe97482600304e352741f384a34f06d5b83ae4

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Hyperbolic and Parabolic Unimodular Random Maps (2016, result): “eral large classes of graphs: The following was proven by the second author [47, 46]. See also earlier works by Pemantle [64], Benjamini, Lyons, Peres, and Schramm [13], and Aldous and Lyons [2]. See [59] for further one-endedness results in the deterministic setting. Theorem 5.15 ([47, 46]). Let (G”
A note on percolation on $Z^d$: isoperimetric profile via exponential cluster repulsion (2007, result): “ome function ψthat satisﬁes X∞ k=1 ψ(k)−2 <∞, (1.4) then the graph contains a transient subtree, and so is transient itself. In particular, IP∗ 2+ε suﬃces for transience. Lyons, Morris and Schramm [LyMS] recently found a very nice few line proof of a reﬁnement of Thomassen’s result, resembling a ”