PreprintCC BY 4.0arXiv
Random walk on the small-world network model in 3 or more dimensions
Abstract
We study the mixing time of a simple random walk on the small-world network defined by adding edges to ℤnd as follows: for each pair {x,y} we add an edge with probability Zn/|x−y|d with Zn chosen so that the average number of added edges to every vertex is $1$. When d≥ 3, we show that with high probability the mixing time is of order log n and that the random walk does not exhibit cutoff.
Related
- A shape theorem for a radially excited random walk in dimensions d >= 2
- Fractal dimensions of paths on three-dimensional random-walk traces
- Renormalizing small ball events for branching random walk
- On the dichotomy of p-walk dimensions on metric measure spaces II
- On the critical branching random walk I: branching capacity and visiting probability
BibTeX
@misc{baran2026random,
title = {{Random walk on the small-world network model in 3 or more dimensions}},
author = {Zsuzsanna Baran and Jonathan Hermon and Anđela Šarković and Allan Sly and Perla Sousi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.12056},
eprint = {2609.12056},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12056},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Random walk on the small-world network model in 3 or more dimensions
AU - Zsuzsanna Baran
AU - Jonathan Hermon
AU - Anđela Šarković
AU - Allan Sly
AU - Perla Sousi
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2609.12056
UR - https://arxiv.org/abs/2609.12056
AB - We study the mixing time of a simple random walk on the small-world network defined by adding edges to ℤ_n^d as follows: for each pair {x,y} we add an edge with probability Z_n/|x−y|^d with Z_n chosen so that the average number of added edges to every vertex is $1$. When d≥ 3, we show that with high probability the mixing time is of order log n and that the random walk does not exhibit cutoff.
ER -