The ASEP speed process

TL;DR

This paper proves that in ASEP with step initial data and a second class particle, the particle's velocity converges to a uniform distribution on [-1,1], confirming a longstanding conjecture and enabling the construction of the ASEP speed process.

Contribution

It establishes the almost sure uniform distribution of the second class particle's velocity in ASEP, resolving key conjectures and constructing the ASEP speed process.

Findings

Second class particle velocity is uniformly distributed on [-1,1] as time approaches infinity.

Resolved Conjecture 1.9 and 1.10 from prior work.

Constructed the ASEP speed process based on the velocity distribution.

Abstract

For ASEP with step initial data and a second class particle started at the origin we prove that as time goes to infinity the second class particle almost surely achieves a velocity that is uniformly distributed on $[-1,1]$. This positively resolves Conjecture 1.9 and 1.10 of [Amir, Angel and Valko, "The TASEP speed process", Annals of Probability 39, 1205--1242, 2011] and allows us to construct the ASEP speed process.