arXivDaily arXiv每日学术速递 周一至周五更新
北美退伍军人节（Veterans Day），12 日无更新
arXiv 0705.0867math.PRmath.CO
收藏

非回溯随机游动的Poisson逼近

Poisson approximation for non-backtracking random walks

Noga Alon, Eyal Lubetzky

备注：
19 pages
阅读 PDF
arXiv 页面 ↗HTML 原文 ↗

摘要：

对扩张图上的随机游动进行了深入的研究，其重要动机是，在某些自然条件下，这些游动可以快速混合，并提供了一种有效的顶点采样方法. Alon、Benjamini、Lubetzky和Sodin研究了正则图上的非回溯随机游动，并证明了它们的混合速率可能是简单随机游动的两倍。作为一个应用，他们证明了在高围长$n$ -顶点正则扩展器上进行长度为$n$的非回溯随机游走，访问顶点的最大次数通常是$(1+o(1))\frac{\log n}{\log\log n}$，就像球和箱实验的情况一样。他们还问，是否可以确定这种散步的确切分布情况。在这项工作中，我们回答了上述问题相结合的一个广义形式的布伦筛与一些扩展的想法在阿隆等人。让$N_t$表示访问的顶点数精确$t$倍的非回溯随机游走长度$n$定期$n$ -顶点扩展的固定度和周长$g$。我们证明，如果$g=ω(1)$，那么对于任何固定的$t$，$N_t/n$通常是$\frac{1}{\mathrm{e}t!}+o(1)$。此外，如果$g=Ω(\log\log n)$，则$N_t/n$通常在所有$t \leq (1-o(1))\frac{\log n}{\log\log n}$上均为$\frac{1+o(1)}{\mathrm{e}t!}$，并且对于所有$t \geq (1+o(1))\frac{\log n}{\log\log n}$为0。特别是，我们得到了上述结果的典型的最大访问数的一个单一的顶点，与改进的阈值窗口。证明的本质在于表明，变量计数访问一组足够远的顶点是渐近独立的泊松变量。

英文摘要：

Random walks on expander graphs were thoroughly studied, with the important motivation that, under some natural conditions, these walks mix quickly and provide an efficient method of sampling the vertices of a graph. Alon, Benjamini, Lubetzky and Sodin studied non-backtracking random walks on regular graphs, and showed that their mixing rate may be up to twice as fast as that of the simple random walk. As an application, they showed that the maximal number of visits to a vertex, made by a non-backtracking random walk of length $n$ on a high-girth $n$-vertex regular expander, is typically $(1+o(1))\frac{\log n}{\log\log n}$, as in the case of the balls and bins experiment. They further asked whether one can establish the precise distribution of the visits such a walk makes. In this work, we answer the above question by combining a generalized form of Brun's sieve with some extensions of the ideas in Alon et al. Let $N_t$ denote the number of vertices visited precisely $t$ times by a non-backtracking random walk of length $n$ on a regular $n$-vertex expander of fixed degree and girth $g$. We prove that if $g=ω(1)$, then for any fixed $t$, $N_t/n$ is typically $\frac{1}{\mathrm{e}t!}+o(1)$. Furthermore, if $g=Ω(\log\log n)$, then $N_t/n$ is typically $\frac{1+o(1)}{\mathrm{e}t!}$ uniformly on all $t \leq (1-o(1))\frac{\log n}{\log\log n}$ and 0 for all $t \geq (1+o(1))\frac{\log n}{\log\log n}$. In particular, we obtain the above result on the typical maximal number of visits to a single vertex, with an improved threshold window. The essence of the proof lies in showing that variables counting the number of visits to a set of sufficiently distant vertices are asymptotically independent Poisson variables.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

↑