Harmonic growth on groups and stationary random graphs

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.