Asymptotic behavior of the Cheeger constant of super-critical percolation in the square lattice

Date

November 6, 2012

Overview

Isoperimetry is a well-studied subject that have found many applications in geometric measure theory (e.g. concentration of measure, heat-kernal estimates, mixing time, etc.) Consider the super-critical bond percolation on mathbb Zd (the d-dimensional square lattice), and φn the Cheeger constant of the super-critical percolation cluster restricted to the finite box [-n,n]d. Following several papers that proved that the leading order asymptotics of φn is of the order 1/n, Benjamini conjectured a limit to n exists. As a step towards this goal, Rosenthal and myself have recently shown that Var(nφn) C n2-d. This implies concentration of n around its mean for dimensions d2. Consider the super-critical bond percolation on mathbb Z2 (the square lattice). We prove the Cheeger constant of the super-critical percolation cluster restricted to finite boxes scale a.s to a deterministic quantity. This quantity is given by the solution to the isoperimetric problem on mathbb R2 with respect to a specific norm. The unique set which gives the solution, is the normalized Wulff shape for the same norm.

Joint work with Marek Biskup, Oren Louidor and Ron Rosenthal.

Speakers

Eviatar Procaccia

Eviatar Procaccia is a Ph.D student at the Weizmann institute of science, advised by Itai Benjamini and Noam Berger. See https://sites.google.com/site/ebprocaccia/ for more details.

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