Pinning of a random walk by a random walk: proof of a conjecture

TL;DR

This paper proves a conjecture regarding the relationship between two radii of convergence related to collision local times of two independent random walks, revealing an intermediate phase in related stochastic models.

Contribution

It settles the full conjecture by comparing variational formulas for the radii, extending previous results to all cases under weak regularity assumptions.

Findings

Proves z1 > z2 for all cases under weak regularity.

Establishes an upper bound for z1 with additional assumptions.

Shows the gap persists with exponential waiting times.

Abstract

In [3] the radius of convergence of the generating function of the collision local time of two independent copies of an irreducible, symmetric and transient random walk on Zd, d \geq 1, was studied. Two versions were considered: z1, the radius of convergence when one walk is averaged over and the other is kept fixed; z2, the radius of convergence when both walks are averaged over. While z2 can be easily computed, no explicit expression is available for z1. In [3] it was conjectured that z1 > z2 under a weak regularity assumption on the random walk. However, this gap was only proved for strongly transient random walk. Subsequently, in [1], [4] and [5] the gap was proved for a subclass of random walks that are transient but not strongly transient. In the present note we settle the full conjecture. The proof is based on a comparison of variational formulas for z1 and z2 derived in [3]. Under additional weak regularity assumptions on the random walk, we derive an upper bound for z1. We further show that the gap persists when the random walk has exponential waiting times. The gaps imply the presence of an intermediate phase for a model of two random walks with a pinning interaction. As explained in [3], they also imply the presence of intermediate phases in three classes of interacting stochastic systems: coupled branching processes, interacting diffusions, and directed polymers with bulk disorder.