On the convoy of the ASEP speed process

TL;DR

This paper analyzes the convoy size in the multi-species ASEP speed process, providing exact formulas, asymptotic behavior, and a critical scaling regime using combinatorial and orthogonal polynomial techniques.

Contribution

It introduces an exact formula for the expected convoy size, proves its universality, and identifies a critical scaling regime for the ASEP speed process.

Findings

Exact formula for expected convoy size

Asymptotic universality for fixed jump rates

Critical scaling regime at q=1 - gamma/√n

Abstract

We investigate the size of the convoy in the speed process in the multi-species asymmetric simple exclusion process (ASEP). Through a coupling argument, we obtain an exact formula for the expected convoy size by relating it to a combinatorial structure. We prove that the asymptotic expected convoy size is universal for all fixed jump rates $q \in [0,1)$. In the special case $q=0$, we upgrade this to full convergence in distribution. We further establish a critical scaling $q = 1 - \gamma/\sqrt{n}$ that yields a nontrivial limiting regime. Our analysis builds on Martin's construction of the convoy and makes use of an orthogonal-polynomial representation of random-walk transition probabilities.