Yuan Tian

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.

In this paper, we study the oriented swap process on the positive integers and its asymptotic properties. Our results extend a theorem by Angel, Holroyd, and Romik regarding the trajectories of particles in the finite oriented swap process. Furthermore, we study the evolution ofthe type of a particle at the leftmost position over time. Our approach relies on a relationship between multi-species particle systems and Hecke algebras, complemented by a detailed asymptotic analysis.