Collisions in the discrete-time TASEP speed process (proof of claimed result)

Prove the claimed collision property in the discrete‑time TASEP speed process studied by Martin and Schmidt (2019), namely that particles with the same asymptotic speed eventually collide. Establishing this result would make the strict upper bound C2 < 1 in Theorem 2.12 on the number of branches along the vertical axis unconditional.

Background

The logarithmic upper bound on the number of branch points of semi‑infinite geodesics along the vertical axis (Theorem 2.12) is shown with a strict constant C2<1 under a claim from the discrete‑time TASEP literature. Specifically, the argument relies on a collision property in the discrete‑time TASEP speed process treated in Martin and Schmidt (2019).

Because that claim lacks a published proof, the paper’s bound is presented conditionally; without it the upper bound weakens to C2≤1. A rigorous proof of the collision property would resolve this dependency and strengthen the branching bound.