TASEP with a moving wall

TL;DR

This paper studies a TASEP with a moving wall, showing it can produce a wide range of fluctuation distributions including Tracy-Widom types, and analyzes the behavior of second class particles in shock scenarios.

Contribution

It introduces a TASEP model with a moving wall that induces diverse fluctuation distributions, extending understanding of particle behavior under dynamic boundary conditions.

Findings

The model produces fluctuation distributions of the form involving Airy processes and arbitrary barriers.

Examples include Tracy-Widom GOE and GUE distributions and their crossover.

The distribution of the second class particle can be a mixture of uniform and atomic measures.

Abstract

We consider a totally asymmetric simple exclusion on $\mathbb{Z}$ with the step initial condition, under the additional restriction that the first particle cannot cross a deterministally moving wall. We prove that such a wall may induce asymptotic fluctuation distributions of particle positions of the form $$ \mathbb{P}\Big(\sup_{\tau\in \mathbb{R}}\{\textrm{Airy}_2(\tau) -g(\tau)\}\leq S\Big)$$ with arbitrary barrier functions $g$. This is the same class of distributions that arises as one-point asymptotic fluctuations of TASEPs with arbitrary initial conditions. Examples include Tracy-Widom GOE and GUE distributions, as well as a crossover between them, all arising from various particles behind a linearly moving wall. We also prove that if the right-most particle is second class, and a linearly moving wall is shock-inducing, then the asymptotic distribution of the position of the second class particle is a mixture of the uniform distribution on a segment and the atomic measure at its right end.