Phase transition on the fluctuation of the structure of random walk ranges
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.
Fields of Study & Subjects
Stochastic processes and statistical mechanics
Geometric Analysis and Curvature Flows
Theoretical and Computational Physics
Publication details
- Journal / source
- arXiv (Cornell University)
- Publication date
- 7 February 2026
- Language
- EN
- Document type
- Preprint