
Speaker Eviatar Procaccia Affiliation Weizmann Institute of Science Host Yuval Peres Duration 00:39:16 Date recorded 6 November 2012 Isoperimetry is a wellstudied subject that have found many applications in geometric measure theory (e.g. concentration of measure, heatkernal estimates, mixing time, etc.) Consider the supercritical bond percolation on mathbb Z^{d} (the ddimensional square lattice), and φ_{n} the Cheeger constant of the supercritical percolation cluster restricted to the finite box [n,n]^{d}. Following several papers that proved that the leading order asymptotics of φ_{n} is of the order 1/n, Benjamini conjectured a limit to nφ_{n} exists. As a step towards this goal, Rosenthal and myself have recently shown that Var(nφ_{n}) C n^{2d}. This implies concentration of nφ_{n} around its mean for dimensions d2. Consider the supercritical bond percolation on mathbb Z^{2} (the square lattice). We prove the Cheeger constant of the supercritical 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.
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