EVENT DETAILS AND ABSTRACT

Title: The chemical distance in critical percolation
Speaker: Michael Damron
Speaker Info: Indiana University
Brief Description:
Special Note:
Abstract:

In two-dimensional critical percolation, the works of Aizenman-Burchard and Kesten-Zhang imply that macroscopic distances inside percolation clusters are bounded below by a power of the Euclidean distance greater than \(1 + \varepsilon\), for some positive epsilon. No more precise lower bound has been given since 1999. Conditional 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 Zhang and Morrow to have volume \(n^{4/3+o(1)}\) on the triangular lattice.

Addressing a question of Kesten and Zhang from 1992, we compare the length of the shortest circuit around 0 in an annulus to that of the innermost circuit (defined analogously to the lowest crossing). The main theorem shows that the ratio of the expected length of the shortest circuit to the expected length of the innermost circuit goes to 0 with \(n\).

This is nearly completed work with Jack Hanson and Phil Sosoe.