正则图上随机游走的最大松弛时间
The maximum relaxation time of a random walk on regular graphs
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中文总结 AI 辅助
本文确定了给定阶数连通简单正则图上随机游走最大松弛时间的渐近精确值，证明了Aldous--Fill谱隙猜想，并给出了极值图的唯一性与稳定性刻画。
中文摘要 AI 辅助
我们确定了给定阶数的连通简单正则图上简单随机游走的渐近最大松弛时间。领先常数取决于阶数的奇偶性：三次图在偶数阶时渐近地达到极值，而四次图在奇数阶时渐近地达到极值。我们得到了一个关于度数一致成立的渐近精确上界；这证明了长期存在的Aldous--Fill谱隙猜想。对非三次图的一致严格改进意味着，对于每一个足够大的偶数阶，最大值由代数连通度最小的三次图唯一取得。我们还证明了三次图近极值者的定量稳定性。对于给定的边连通度，当度数趋于无穷时，我们确定了精确界并刻画了渐近等号成立的情形。
英文摘要
We establish sharp quadratic bounds on the relaxation time of simple random walk on connected regular graphs, with leading constants $3/(2π^2)$ and $1/π^2$ for even and odd orders, respectively. This resolves a longstanding conjecture known as Aldous--Fill spectral gap conjecture (2002). We also prove uniqueness and stability theorems for the corresponding cubic and quartic chains, and settle the quartic uniqueness conjecture posed by Abdi, Ghorbani, and Imrich (2021) and by Abdi and Ghorbani (2023). We establish sharp bounds on algebraic connectivity under minimum-degree and regularity constraints and prove the conjectured chain structure of the minimisers, identifying their repeating blocks. This resolves a longstanding conjecture of Guiduli and Mohar (1996) and a structural conjecture of Abdi and Ghorbani (2024). Our quantitative stability theorem shows that nearly minimum algebraic connectivity forces nearly maximum diameter, proving their diameter conjecture. We further obtain sharp relaxation-time bounds in terms of edge-connectivity. For nonregular graphs, we establish sharp bounds for the gap between maximum degree and adjacency spectral radius, confirming a conjecture of Liu (2024). As applications, we prove the sharp bounds on hitting and commute times conjectured by Aldous and Fill (2002) for regular graphs.
