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Percolation Phase Transition via the Gaussian Free Field

Date: Tuesday, May 21, 2019 17:30 - 18:30
Speaker: Aran Raoufi (ETH Zürich)
Location: Big Seminar room Ground floor / Office Bldg West (I21.EG.101)
Series: Mathematics and CS Seminar
Host: M. Beiglböck, N. Berestycki, L. Erdös, J. Maas
Lab building west seminar room

Abstract:

Let $G$ be a bounded-degree infinite graph, and $p_c$ be the critical parameter of bond percolation on $G$. That is $p_c$ is the infimum of values of $p$ that you have an infinite cluster almost surely.

In this talk, we prove that if the isoperimetric dimension of $G$ is higher than 4, then $p_c(G)<1$. The theorem settles affirmatively two conjectures of Benjamini and Schramm. Notably, if $G$ is a transitive graph with super-linear growth, then $p_c(G) <1$. In particular, it implies that if $G$ is a Cayley graph of a finitely generated group without a finite index cyclic subgroup, then $p_c(G)<1$.

The proof of the theorem starts with the existence of an infinite cluster for percolation in a certain in-homogeneous random environment governed by the Gaussian free field. Then, by the help of a multiscale decomposition of GFF, we relate the existence of an infinite cluster in percolation in the random environment to that of percolation with a fix parameter $p<1$.

This talk is based on a joint work with H. Duminil-Copin, S. Goswami, F. Severo, and A. Yadin.
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