Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Author: Benjamini, Itai
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214752091