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arXiv · 2609.06818

The maximum relaxation time of a random walk on regular graphs

Haoran Zhu
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Abstract

We determine the asymptotic maximum relaxation time of simple random walk on connected simple regular graphs of a given order. The leading constant depends on the parity of the order: cubic graphs are asymptotically extremal at even orders, whereas quartic graphs are asymptotically extremal at odd orders. We obtain an asymptotically sharp upper bound uniform in the degree; this proves the longstanding Aldous--Fill spectral gap conjecture. A uniform strict improvement for noncubic graphs implies that, for every sufficiently large even order, the maximum is attained uniquely by the cubic graph of least algebraic connectivity. We also prove quantitative stability for cubic near-extremisers. For prescribed edge-connectivity, we determine the sharp bound when the degree tends to infinity and characterise asymptotic equality.

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Haoran Zhu. 2026-09-06. The maximum relaxation time of a random walk on regular graphs. https://arxiv.org/abs/2609.06818

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