Hao-Ran Zhu

The maximum relaxation time of a random walk on regular graphs

We establish sharp quadratic bounds on the relaxation time of simple random walk on connected regular graphs, with leading constants $3/(2\pi^2)$ and $1/\pi^2$ for even and odd orders, respectively. This resolves a longstanding conjecture known as Aldous--Fill spectral gap conjecture (2002). We also prove uniqueness an...