Work

On the self-intersection time of non-backtracking random walks

Year: 2026

Type: preprint

Abstract: We study the self-intersection time of the non-backtracking random walk on connected undirected graphs. For every fixed $Δ\geq 3$ we show that the expected self-intersection time is $O(\sqrt{n} \log n)$ on $n$-vertex graphs with minimum degree at least $3$ and maximum degree at most $Δ$. For regular graphs with a uniform spectral gap, we improve this to $O(\sqrt{n})$. We also show an $Ω(\sqrt{n})$ lower bound on a class of regular expanders. Our upper bound on the expected self-intersection time implies an improved mixing time bound on Glauber dynamics for the Ising model on $Δ$-regular graphs at the tree uniqueness threshold.

Source: arXiv (Cornell University)

DOI: 10.48550/arxiv.2608.09729

Authors: Bencs, Ferenc, Goldberg, Leslie Ann, Jenssen, Matthew, Jerrum, Mark, Pete, Gabor, Regts, Guus, Yin, Yitong

Institutions: Centrum Wiskunde & Informatica, Nanjing University of Science and Technology, Nanjing University, University of Oxford, King's College London, Queen Mary University of London, University of Amsterdam

Topic: Stochastic processes and statistical mechanics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Nederlandse Organisatie voor Wetenschappelijk Onderzoek, National Research, Development and Innovation Office

OpenAlex ID: W7202218293

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