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Very sharp distance and range transitions for random walk bridges on Ramanujan graphs

  • Itai Benjamini
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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 √(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.

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Cite

BibTeXRIS
BibTeX
@misc{benjamini2026very,
  title = {{Very sharp distance and range transitions for random walk bridges on Ramanujan graphs}},
  author = {Itai Benjamini},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.35213},
  eprint = {2609.35213},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.35213},
  note = {CC0},
}
RIS
TY  - UNPB
TI  - Very sharp distance and range transitions for random walk bridges on Ramanujan graphs
AU  - Itai Benjamini
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.35213
UR  - https://arxiv.org/abs/2609.35213
AB  - 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 √(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.
ER  -