发表机构
Shanghai Institute for Mathematics and Interdisciplinary Sciences (SIMIS); Research Institute of Intelligent Complex Systems, Fudan University（上海数学与交叉学科研究院; 复旦大学智能复杂系统研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究比较LPS图与无限正则树上随机游走的返回概率，利用Ihara zeta函数修正环的影响，在平均步长与最短环长相当的情形下给出上界，并在广义黎曼猜想下得到与顶点数倒数同阶的修正。
AI 中文摘要
我们比较了在Lubotzky--Phillips--Sarnak图与无限正则树上，于独立几何时间停止的随机游走的返回概率，其中McKay测度给出树上的值，Ihara zeta函数描述由环引起的额外返回。当平均游走长度与最短环的长度相当时，我们估计了这一修正，在此情形下，足够长以穿越一个环的游走获得不可忽略的权重。无条件地，我们证明了在乘以顶点数后，素数水平上具有双对数因子和有界固定矩的上界。在二次Dirichlet $L$-函数的广义黎曼猜想下，该修正与顶点数的倒数同阶。
英文摘要
We compare return probabilities for random walks stopped at an independent geometric time on Lubotzky--Phillips--Sarnak graphs and on the infinite regular tree, with the McKay measure giving the tree value and the Ihara zeta function describing the additional returns caused by cycles. We estimate this correction when the mean walk length is comparable to the length of the shortest cycle, a regime in which walks long enough to traverse a cycle receive non-negligible weight. Unconditionally, we prove an upper bound with a double logarithmic factor and bounded fixed moments over prime levels after multiplication by the number of vertices. Under the Generalized Riemann Hypothesis for quadratic Dirichlet $L$-functions, the correction is of the same order as the reciprocal of the number of vertices.
Comments21 pages. Comments welcome
