Work

Random walk on the small-world network model in 3 or more dimensions

Year: 2026

Type: preprint

Abstract: We study the mixing time of a simple random walk on the small-world network defined by adding edges to $\mathbb{Z}_n^d$ 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 $d\geq 3$, we show that with high probability the mixing time is of order $\log n$ and that the random walk does not exhibit cutoff.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2609.12056

Authors: Baran, Zsuzsanna, Hermon, Jonathan, Šarković, Anđela, Sly, Allan, Sousi, Perla

Institutions: Université Paris Dauphine-PSL, University of British Columbia, University of Cambridge, Princeton University

Language: English

Topic: Stochastic processes and statistical mechanics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Natural Sciences and Engineering Research Council of Canada, Engineering and Physical Sciences Research Council

OpenAlex ID: W7212527116

Cites:

Cited by: 1

Related to: 10