6,084 research outputs found
Indistinguishability of Trees in Uniform Spanning Forests
We prove that in both the free and the wired uniform spanning forest (FUSF
and WUSF) of any unimodular random rooted network (in particular, of any Cayley
graph), it is impossible to distinguish the connected components of the forest
from each other by invariantly defined graph properties almost surely. This
confirms a conjecture of Benjamini, Lyons, Peres and Schramm.
We use this to answer positively two additional questions of Benjamini,
Lyons, Peres and Schramm under the assumption of unimodularity. We prove that
on any unimodular random rooted network, the FUSF is either connected or has
infinitely many connected components almost surely, and, if the FUSF and WUSF
are distinct, then every component of the FUSF is transient and
infinitely-ended almost surely. All of these results are new even for Cayley
graphs.Comment: 43 pages, 2 figures. Version 2: minor corrections and improvements;
references added; one additional figur
Counterexamples for percolation on unimodular random graphs
We construct an example of a bounded degree, nonamenable, unimodular random
rooted graph with for Bernoulli bond percolation, as well as an
example of a bounded degree, unimodular random rooted graph with but
with an infinite cluster at criticality. These examples show that two
well-known conjectures of Benjamini and Schramm are false when generalised from
transitive graphs to unimodular random rooted graphs.Comment: 20 pages, 3 figure
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