Ends in Uniform Spanning Forests
Russell Lyons, Benjamin J. Morris, Oded Schramm

TL;DR
This paper extends the understanding of the structure of wired spanning forests by relaxing symmetry assumptions, showing that trees in such forests typically have only one end under broader conditions related to isoperimetric profiles.
Contribution
It introduces a new condition based on isoperimetric profiles that ensures trees in wired spanning forests have only one end, removing the need for transitivity and unimodularity.
Findings
Trees in wired spanning forests have only one end under broader conditions.
The new isoperimetric condition generalizes previous symmetry assumptions.
The results apply to a wider class of graphs beyond transitive unimodular ones.
Abstract
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.
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Taxonomy
TopicsGeometric and Algebraic Topology · Stochastic processes and statistical mechanics · Advanced Combinatorial Mathematics
