Applied Maths Seminar: Parkinson -- Random walks on homogeneous trees

James Parkinson, School of Mathematics and Statistics, The University of Sydney 

Wednesday 12th May 14:05-14:55pm, New Law School Seminar 030 (Building F10).  

"Trees" are ubiquitous in pure and applied mathematics, arising in such diverse areas as
the study of branching processes and the representation theory of the special linear
group SL_2 over a p-adic field.  A homogeneous tree is a tree in which each vertex has
the same valency.  A random walk on the vertices of a homogeneous tree is called
isotropic if the transition probabilities p(x,y) of the walk only depend on the graph
distance between x and y.  In this talk we outline how some basic harmonic analysis can
be used to derive a precise asymptotic formula for the n-step transition probabilities
p_n(x,y) of an isotropic random walk as n approaches infinity, with x and y fixed
vertices (ie, a local limit theorem).  

This work is classical, dating back (at least) to S.  Sawyer in 1978.  But the technique
is rather robust: Recently it has been vastly generalised to study random walks on
"affine buildings" by Cartwright, Woess, Parkinson, Shapira and others, thereby giving
local limit theorems on semi-simple Lie groups over p-adic fields.