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We construct an example of a bounded degree, nonamenable, unimodular randomrooted graph with $p_c=p_u$ for Bernoulli bond percolation, as well as anexample of a bounded degree, unimodular random rooted graph with $p_c<1$ butwith an infinite cluster at criticality. These examples show that twowell-known conjectures of Benjamini and Schramm are false when generalised fromtransitive graphs to unimodular random rooted graphs.

Counterexamples for percolation on unimodular random graphs