AI 中文总结
该研究分析了连通无向图上非回溯随机游走的自相交时间，得出不同条件下的上下界，并将其应用于改进Δ正则图上Ising模型Glauber动力学的混合时间界。
AI 中文摘要
我们研究连通无向图上非回溯随机游走的自相交时间。对每个固定的Δ≥3，我们证明在最小度至少为3、最大度至多为Δ的n顶点图上，期望自相交时间为O(√n log n)；对于具有均匀谱间隙的正则图，我们将其改进为O(√n)；我们还在一类正则扩展器上证明了Ω(√n)的下界。我们得到的期望自相交时间上界，意味着在树唯一性阈值下，Δ正则图上Ising模型的Glauber动力学混合时间界得到了改进。
英文摘要
We study the self-intersection time of the non-backtracking random walk on connected undirected graphs. For every fixed $Δ\geq 3$ we show that the expected self-intersection time is $O(\sqrt{n} \log n)$ on $n$-vertex graphs with minimum degree at least $3$ and maximum degree at most $Δ$. For regular graphs with a uniform spectral gap, we improve this to $O(\sqrt{n})$. We also show an $Ω(\sqrt{n})$ lower bound on a class of regular expanders. Our upper bound on the expected self-intersection time implies an improved mixing time bound on Glauber dynamics for the Ising model on $Δ$-regular graphs at the tree uniqueness threshold.
