On the chemical distance in critical percolation

Stochastics Seminar
Thursday, September 3, 2015 - 3:05pm
1 hour (actually 50 minutes)
Skiles 006
School of Mathematics, Georgia Tech
In two-dimensional critical percolation, the work of Aizenman-Burchard implies that macroscopic distances inside percolation clusters are bounded below by a power of the Euclidean distance greater than 1+\epsilon, for some positive \epsilon. No more precise lower bound has been given so far. Conditioned on the existence of an open crossing of a box of side length n, there is a distinguished open path which can be characterized in terms of arm exponents: the lowest open path crossing the box. This clearly gives an upper bound for the shortest path. The lowest crossing was shown by Morrow and Zhang to have volume n^4/3 on the triangular lattice. In 1992, Kesten and Zhang asked how, given the existence of an open crossing, the length of the shortest open crossing compares to that of the lowest; in particular, whether the ratio of these lengths tends to zero in probability. We answer this question positively.