For every infinite connected locally finite nonamenable quasi-transitive graph under Bernoulli bond percolation, p_c<p_{2→2}≤p_u, yielding a nonempty parameter interval with infinitely many infinite clusters almost surely.
For every infinite connected locally finite nonamenable quasi-transitive graph under Bernoulli bond percolation, p_c<p_{2→2}≤p_u, yielding a nonempty parameter interval with infinitely many infinite clusters almost surely.
For every infinite connected locally finite nonamenable quasi-transitive graph under Bernoulli bond percolation, p_c<p_{2→2}≤p_u, yielding a nonempty parameter interval with infinitely many infinite clusters almost surely.
If correct, resolves both the nonuniqueness conjecture and the stronger operator-threshold question in this graph class.
Must nonamenable quasi-transitive graphs have a nonempty bond-percolation nonuniqueness phase?
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OpenAI reports nonuniqueness of percolation on nonamenable quasi-transitive graphs
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