Leaders in multi-type TASEP

TL;DR

This work analyzes a multi-type TASEP on $\mathbb{Z}$ with step initial data, focusing on the leader—the rightmost particle's type—and establishes a central limit theorem for $L_1(t)/\sqrt{t}$ with a half-normal limit. It derives a comprehensive integral representation for leader-related observables via a colored TASEP/colored vertex-model duality, and connects these to voter and coalescence processes through a color-position duality. The paper also proves joint-distribution results for multiple leaders, calculates the asymptotics for leader-type changes, and introduces the TASEP ranking process, unveiling dualities that translate leader statistics into ranking dynamics. Overall, the results illuminate the interplay between multi-type TASEP, vertex-model observables, and classical interacting particle systems, with explicit asymptotics and integral formulas enabling precise probabilistic descriptions of leader behavior. The methods provide a framework to study refined observables in integrable particle systems and their connections to voter/coalescence dynamics, offering new probabilistic insights and potential extensions to related multi-species models.

Abstract

We study the totally asymmetric simple exclusion process (TASEP) on $\mathbb{Z}$ with step initial condition, in which all particles have distinct types. Our main object of interest is the type of the rightmost particle -- the leader -- at large time $t$. We prove a central limit theorem for this random variable. Somewhat unexpectedly, the problem is closely connected to certain observables of voter and coalescing processes on $\mathbb{Z}$; we therefore derive their asymptotics as well. We also analyze the large-time behavior of a few other related observables, including certain multi-particle ones.

Figures (2)

• Figure 1: TASEP with the multi-type step initial conditon (top) and a possible configuration of the TASEP after several jumps (bottom).
• Figure 2: Solid line: $\Re \left( \frac{w^2}{1+w} \right)=0$, the regions of positive and negative real part of this function are also depicted. We deform (without encountering any poles) the contour of integration to the dashed line, which lies almost entirely in the region with negarive real part, and only in an (infinitesimal, of order $t^{-1/2}$) neighborhood of $w=0$ the values of $\Re \left( \frac{w^2}{1+w} \right)$ on the contour are close to 0.

Theorems & Definitions (44)

• Theorem 1.1: Theorem \ref{['th:leader-type']} below
• Theorem 1.2: Theorem \ref{['th:leader-changes']} below
• Theorem 2.1
• Corollary 2.2
• Proposition 2.3
• Proposition 2.4
• proof : Proof of Proposition \ref{['prop:duality']}
• proof : Proof of Proposition \ref{['prop:TASEP-transition']}
• Example 4.1
• Proposition 4.2