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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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