Phase transition on the fluctuation of the structure of random walk ranges

Arka Adhikari; Izumi Okada

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arXiv · 2602.07304

Phase transition on the fluctuation of the structure of random walk ranges

Arka Adhikari·Izumi Okada

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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.

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Arka Adhikari, Izumi Okada. 2026-02-07. Phase transition on the fluctuation of the structure of random walk ranges. https://arxiv.org/abs/2602.07304

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