Ramanujan 图上随机游走桥的非常尖锐的距离与范围转变
Very sharp distance and range transitions for random walk bridges on Ramanujan graphs
AI总结：
本文证明在具有对数围长的顶点传递 Ramanujan 图上，长度为对数阶的随机游走桥的最大距离在临界窗口内从根号对数阶突变至对数阶，并给出范围的两点极限定律。
AI中文摘要：
对于具有对数围长的顶点传递 Ramanujan 图，长度为 $\log N$ 阶的简单随机游走桥（其中 $N$ 为图的大小）的最大距离在有界临界窗口内从 $\sqrt{\log N}$ 阶变为 $\log N$ 阶。我们通过区分其提升到正则树后闭合的桥与不闭合的桥来证明这一点。一个一致的两项返回估计决定了这两种情况的概率以及实值临界中心。在相同的 $O(1)$ 窗口内，归一化范围具有一个两点极限定律，其混合权重在窗口内非平凡地变化。
英文摘要：
For vertex-transitive Ramanujan graphs with logarithmic girth, a simple random walk bridge of length of order $\log N$, where $N$ is the size of the graph, has a maximum distance that changes from order $\sqrt{\log N}$ to order $\log N$ in a bounded critical window. We prove this by separating bridges whose lifts to the regular tree close from those whose lifts do not. A uniform two-term return estimate determines the probabilities of these two cases and the real-valued critical center. In the same $O(1)$ window, the normalized range has a two-point limiting law whose mixture weights vary nontrivially across the window.
