Boundary case for collisions at the critical tail exponent

Determine whether, for an i.i.d. random tooth profile satisfying $P(H_0>z)\sim Cz^{-1/3}$ as $z\to\infty$, a typical realization of the random comb graph $\mathrm{Comb}(\mathbb{Z},H^\omega)$ has the infinite collision property for two independent simple random walks; in particular, establish or refute the conjecture that this critical case belongs to the infinite-collision regime.

Background

The paper proves a phase transition for two-walk collisions on random comb graphs with i.i.d. tooth lengths whose tail satisfies $P(H_0>z)\sim Cz^{-\gamma}$: typical realizations have infinitely many collisions when $\gamma>1/3$ and finitely many collisions when $\gamma<1/3$. The critical value $\gamma=1/3$ is not covered by the analysis.

The authors explain that the proof for $\gamma>1/3$ relies on uniform heat-kernel estimates and a second-moment argument over large disks. The technical limitations of those estimates leave the boundary case unresolved, although the authors conjecture that it should have the same infinite-collision behavior as the supercritical regime.