Abstract:
I will discuss the number of infinite clusters in the Bernoulli bond percolation of a finitely generated Cayley graph of a group. Except for a few cases, we do not know the number of infinite clusters at the uniqueness threshold, and there is not a general conjecture either. We show that if the group has an amenable, normal, subgroup with an exponential growth rate, then the Bernoulli bond percolation has a non-unique infinite cluster. This covers interesting cases like the lamplighter group over a nonamenable group.
The main tool we develop is called “subgroup relativization”, and its first application is on the study the intersection of an independent random walk on the Cayley graph with an infinite cluster, where we solved a conjecture by Lyons and Schramm (1999). The second application is about percolation on 
, the direct product of a regular tree with a lattice. We show that the Hausdorff dimension of the set of accumulation points of each infinite cluster in the boundary of the tree has a jump discontinuity from at most 1/2 to 1 at the uniqueness threshold 
.
This is joint work with Tom Hutchcroft.
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