Abstract: 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.

Another question we address is how small can harmonic growth of groups be? We are able to construct a group with a logarithmic harmonic growth. We also have an argument why certain naive approaches cannot construct smaller than logarithmic harmonic growth.

No prior knowledge is assumed, and we will define everything in the talk. Joint work with Itai Benjamini, Hugo Duminil-Copin and Gady Kozma.

Coffee will be served at 12:00 before the lecture
at Schreiber building 006