Problem
Research questions and friction points this paper is trying to address.
non-backtracking random walk
self-intersection time
graph
regular expander
spectral gap
On the self-intersection time of non-backtracking random walks
🤖 AI Summary
This study investigates the self-intersection time of non-backtracking random walks on connected undirected graphs. For $n$-vertex graphs with minimum degree at least 3 and maximum degree at most $\Delta$, the authors combine spectral graph theory, probabilistic analysis, and structural properties of paths to establish, for the first time, an upper bound of $O(\sqrt{n} \log n)$ on the expected self-intersection time over general graph families. On regular graphs with a uniform spectral gap, this bound is further improved to $O(\sqrt{n})$, and a matching lower bound of $\Omega(\sqrt{n})$ is proven. These results directly yield an improved mixing time bound for Glauber dynamics of the Ising model below the tree uniqueness threshold, highlighting the role of the non-backtracking mechanism in accelerating convergence.
📝 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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
non-backtracking random walk
self-intersection time
spectral gap
regular expander
mixing time
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