The Best Mixing Time for Random Walks on Trees

We characterize the extremal structures for mixing walks on trees that start from the most advantageous vertex. Let $G=(V,E)$ be a tree with stationary distribution $π$. For a vertex $v \in V$, let $H(v,π)$ denote the expected length of an optimal stopping rule from $v$ to $π$. The \emph{best mixing time} for $G$ is $\min_{v \in V} H(v,π)$. We show that among all trees with $|V|=n$, the best mixing time is minimized uniquely by the star. For even $n$, the best mixing time is maximized by the uniquely path. Surprising, for odd $n$, the best mixing time is maximized uniquely by a path of length $n-1$ with a single leaf adjacent to one central vertex.

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