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

Friday 20th March 2015 - 14:00 to 15:00
INI Seminar Room 1
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.

University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons