Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the convoy of the ASEP speed process

Yuan Tian

TL;DR

This work analyzes the convoy size in the ASEP speed process, connecting probabilistic convoy structure to an exact formula via $q$-Genocchi numbers and to a universal $\sqrt{n}$-scaling for fixed $q\in[0,1)$. It leverages Martin's convoy construction and an orthogonal-polynomial framework built from continuous big $q$-Hermite and Askey–Wilson polynomials, together with a birth-death–Karlin–McGregor representation, to obtain both explicit formulas and limiting results. A notable result is the universality of the normalized convoy size, with $\lim_{n\to\infty}\frac{1}{\sqrt{n}}\mathbb{E}_x[\#\mathcal C_0^n]=\sqrt{\frac{4x(1-x)}{\pi}}$, and a distributional limit in the special case $q=0$; a critical weakly asymmetric scaling $q_n=e^{-\gamma/\sqrt{n}}$ yields a nontrivial limiting regime described by a joint density $f^{\gamma}_{X,Y}$.

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 $q = 1 - γ/\sqrt{n}$ 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.

On the convoy of the ASEP speed process

TL;DR

This work analyzes the convoy size in the ASEP speed process, connecting probabilistic convoy structure to an exact formula via -Genocchi numbers and to a universal -scaling for fixed . It leverages Martin's convoy construction and an orthogonal-polynomial framework built from continuous big -Hermite and Askey–Wilson polynomials, together with a birth-death–Karlin–McGregor representation, to obtain both explicit formulas and limiting results. A notable result is the universality of the normalized convoy size, with , and a distributional limit in the special case ; a critical weakly asymmetric scaling yields a nontrivial limiting regime described by a joint density .

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 . In the special case , we upgrade this to full convergence in distribution. We further establish a critical scaling 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.
Paper Structure (15 sections, 49 theorems, 281 equations, 2 figures)

This paper contains 15 sections, 49 theorems, 281 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Definitions and Main results
  4. Related work
  5. The Martin's Construction
  6. Properties of the inhomogeneous random walk and first applications
  7. Properties of the inhomogeneous random walk
  8. Asymptotic size of the TASEP convoy
  9. An exact formula for the ASEP convoy
  10. Asymptotics size of the ASEP convoy
  11. The continuous big q-Hermite orthogonal polynomials
  12. Birth and death chains and orthogonal polynomials
  13. Proof of Theorem \ref{['thm.3']}
  14. Weakly asymmetric limit of the convoy size
  15. The proof of Theorem \ref{['thm.main']}

Key Result

Theorem 1.4

Let $Y_0\left(t\right)$ denote the position of the second class particle at time $t$ in the ASEP with step initial condition and a single second-class particle. Then where $U$ is a uniform random variable on $\left[-1,1\right]$. We define $U$ to be the speed of the second-class particle.

Figures (2)

  • Figure 1: Size of ASEP convoy $q = 0$
  • Figure 2: $q = 0.5$

Theorems & Definitions (112)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: mountford2005motion, Theorem 1, aggarwal2023asep, Theorem 1.1
  • Definition 1.5: amir2011tasep, Definition 1.3
  • Definition 1.6: amir2011tasep, Theorem 1.8
  • Remark 1.7
  • Theorem 1.8: Theorem \ref{['thm.main']}
  • Theorem 1.9: Theorem \ref{['thm.uni']}
  • Theorem 1.9: Theorem \ref{['thm.uni']}
  • ...and 102 more