Abstract:
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 $\mathbb{Z}^d$ (the d-dimensional square lattice),
and $\phi_n$ 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
$\phi_n$ is of the order $1/n$, Benjamini conjectured a limit to $n\phi_n$ exists. As a step towards
this goal, Rosenthal and myself have recently shown that $Var(n\phi_n)< C n^{2-d}$. This implies
concentration of $n\phi_n$ around its mean for dimensions $d>2$. Consider the super-critical
bond percolation on $\mathbb{Z}^2$ (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 $\mathbb{R}^2$ 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.
Bio:
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.