Infinitely many collisions on random comb graphs at the critical exponent $\gamma=1/3$
Year: 2026
Type: preprint
Abstract: Let $\operatorname{Comb}(\mathbb{Z},H)$ be the comb obtained by attaching to each $x\in\mathbb{Z}$ a tooth of height $H_x$, where the $H_x$ are i.i.d.\ with $\mathbb{P}(H_x>z)\sim Cz^{-\gamma}$. De Ambroggio, Ng, Nitzschner and Scali proved that two independent simple random walks on a typical such comb collide infinitely often almost surely when $\gamma>1/3$ and only finitely often when $\gamma<1/3$, and conjectured infinitely many collisions at $\gamma=1/3$. We prove this conjecture. At the critical exponent the expected number of collisions in one scale is of order one while its second moment is at most of order $\log N$, so the Paley–Zygmund inequality gives a collision in the $m$-th dyadic scale with conditional probability at least $c/m$, and Lévy's conditional Borel–Cantelli lemma applies because $\sum 1/m=\infty$. The main step is an annealed bound, uniform in the scale, on the expected number of collisions at small times, which replaces the polynomial losses in the heat-kernel estimates of the supercritical proof by bounded factors.
Author: Shmalo, Yitzchak
Institution: Hebrew University of Jerusalem
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225810825
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