Phase transition on the fluctuation of the structure of random walk ranges
Probability
2026-03-18 v2
Abstract
We investigate fluctuation phenomena for the graph distance and the number of cut points associated with random media arising from the range of a random walk. Our results demonstrate a sequence of dimension-dependent phase transitions in the scaling behavior of these fluctuations, leading to qualitatively different regimes, with a distinct phase transition in dimension 6. In particular, we remark that convergence in dimension 6 occurs with a non-standard rescaling.
Cite
@article{arxiv.2602.07304,
title = {Phase transition on the fluctuation of the structure of random walk ranges},
author = {Arka Adhikari and Izumi Okada},
journal= {arXiv preprint arXiv:2602.07304},
year = {2026}
}