Mixing time for the random walk on the range of the random walk on tori

Abstract

Consider the subgraph of the discrete $d$-dimensional torus of size length $N$, $d\ge 3$, induced by the range of the simple random walk on the torus run until the time $u{N}^d$. We prove that for all $d\ge 3$ and $u>0$, the mixing time for the random walk on this subgraph is of order $N^2$ with probability at least $1-C{e}^{-(\log N{)}^2}$.

arXiv comment11 pages