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

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.