On the convoy of the ASEP speed process

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.