| 概要 | We consider the unique infinite connected component of supercritical
bond percolation on the square lattice and study the geometric
properties of isoperimetric sets, i.e., sets with minimal boundary for
a given volume. For almost every realization of the infinite connected
component we prove that, as the volume of the isoperimetric set tends
to infinity, its asymptotic shape can be characterized by an
isoperimetric problem in the plane with respect to a particular norm.
As an application we then show that the anchored isoperimetric profile
with respect to a given point as well as the Cheeger constant of the
giant component in finite boxes scale to deterministic quantities.
This settles a conjecture of Itai Benjamini for the plane. Only
elementary knowledge in probability theory will be assumed. Joint work
with M. Biskup, E. Procaccia and R. Rosenthal. |