Technion
 
Mathematics Departement Colloquium
 
Speaker: Ariel Yadin (BGU)
 
Title:   Harmonic growth on groups and stationary random graphs  
 
Time:    Monday, November 7, 3:30 pm
 
Place:   Amado 232
 
Colloquim website:  < http://www.technion.ac.il/~yehuday/colloquium/colloq.html>
 
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.
 
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Technion Math. Net (TECHMATH)
Editor: Michael Cwikel   <techm@math.technion.ac.il> 
Announcement from: Amir Yehudayoff   <amir.yehudayoff@gmail.com>