We use cookies to ensure that we give you the best experience on our website. You can change your cookie settings at any time. Otherwise, we'll assume you're OK to continue.

Department of Mathematical Sciences

# Seminar Archives

## Statistics Seminars: Uniformity of the late points of random walk

Presented by Perla Sousi, University of Cambridge

24 November 2014 14:00 in CM221

Let X be a simple random walk in Z dn and let t cov be the expected amount of time it
takes for X to visit all of the vertices of Z dn . For α ∈ (0, 1), the set L α of α-late points consists of
those x ∈ Z dn which are visited for the first time by X after time αt cov . Oliveira and Prata (2011)
showed that the distribution of L 1 is close in total variation to a uniformly random set. The value
α = 1 is special, because |L 1 | is of order 1 uniformly in n, while for α < 1 the size of L α is of
order n d−αd . In joint work with Jason Miller we study the structure of L α for values of α < 1.
In particular we show that there exist α 0 < α 1 ∈ (0, 1) such that for all α > α 1 the set L α looks
uniformly random, while for α < α 0 it does not (in the total variation sense). In this talk I will try
to explain the main ideas of our proof and what are the next steps in this direction.