Percolation Today is an online seminar aiming to promote the exchange of exciting new ideas on both Bernoulli and dependent percolation processes, at a level appropriate for specialists in the area. We hope to be a focal point to foster new connections and collaborations between the global community of percolation lovers.
You are warmly invited to sign up to our mailing list and attend to our seminars via Zoom.
Infinite clusters at the uniqueness threshold and subgroup relativization
Speakers: Minghao Pan
Date: March, 3rd, 16:00 CET
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.
If you already signed to our newsletter, you will receive an individual invitation. Otherwise you can register yourself here.
Organizers:
Barbara Dembin
Piet Lammers
Hugo Vanneuville
Franco Severo
Scientific Commitee:
Christophe Garban
Tom Hutchcroft
Pictures by: Adèle Ivain and Laurie Quétel