Harmonic growth on groups and stationary random graphs
Ariel Yadin, Ben Gurion University
13/05/2012 - 17:00
We study the structure of harmonic functions on certain homogeneous graphs: Cayley graphs, and stationary random graphs which are homogeneous "on average". Harmonic functions have been used to understand the geometry of these objects. Two notable examples are: Kleiner's proof of Gromov's theorem regarding polynomial volume growth groups, and the invariance principle (CLT) for super-critical percolation clusters on Z^d.
We consider the following question: What is the minimal growth of a non-constant harmonic function on a graph G (as above)?
This question is of course closely related to the Liouville property and Poisson-Furstenberg boundary; a non-Liouville graph just means that there exist _bounded_ non-constant harmonic functions. A classical result of Kaimanovich & Vershik relates the Liouville property on groups to sublinear entropy of the random walk.
We show a simple but very useful inequality regarding harmonic functions and entropy on a group. This inequality allows us to deduce many results rather simply, among them:
1. A quantified version of one direction of Kaimanovich & Vershik
2. Groups (and stationary graphs) of polynomial growth have no sub-linear non-constant harmonic functions.
3. Uniqueness of the "corrector" for super-critical percolation on Z^d.
All notions will be defined during the talk.
תאריך עדכון אחרון : 08/05/2012