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