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

TL;DR

This paper studies how the fluctuations in the structure of random walk ranges change with dimension, revealing phase transitions in their scaling behavior, especially notable at dimension 6.

Contribution

It uncovers dimension-dependent phase transitions in the fluctuation behavior of random walk ranges, including a unique transition at dimension 6 with non-standard rescaling.

Findings

Dimension-dependent phase transitions in fluctuation behavior

Distinct regimes identified for different dimensions

Unique convergence behavior at dimension 6

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.