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Uniformity of the late points of random walk on $\mathbb{Z}_d^n$ for $d \geq 3$

Date: 
Friday 20th March 2015 - 14:00 to 15:00
Venue: 
INI Seminar Room 1
Abstract: 
Co-author: Jason Miller (MIT)

Let $X$ be a simple random walk in $\mathbb{Z}_n^d$ and let $t_{\rm{cov}}$ be the expected amount of time it takes for $X$ to visit all of the vertices of $\mathbb{Z}_n^d$. For $\alpha\in (0,1)$, the set $\mathcal{L}_\alpha$ of $\alpha$-late points consists of those $x\in \mathbb{Z}_n^d$ which are visited for the first time by $X$ after time $\alpha t_{\rm{cov}}$. Oliveira and Prata (2011) showed that the distribution of $\mathcal{L}_1$ is close in total variation to a uniformly random set. The value $\alpha=1$ is special, because $|\mathcal{L}_1|$ is of order 1 uniformly in $n$, while for $\alpha<1$ the size of $\mathcal{L}_\alpha$ is of order $n^{d-\alpha d}$. In joint work with Jason Miller we study the structure of $\mathcal{L}_\alpha$ for values of $\alpha<1$. In particular we show that there exist $\alpha_0<\alpha_1 \in(0,1)$ such that for all $\alpha>\alpha_1$ the set $\mathcal{L}_\alpha$ looks uniformly random, while for $\alpha<\alpha_0$ 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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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons