Shock fluctuations in flat TASEP under critical scaling

TL;DR

This paper investigates the behavior of shock fluctuations in a TASEP model under a critical scaling regime, revealing new limiting distributions and their convergence properties, which extend understanding of phase transitions in particle systems.

Contribution

It introduces the critical scaling $1- ext{alpha} = a t^{-1/3}$ in TASEP, analyzing the resulting shock fluctuations and their limiting distributions, which was not previously studied.

Findings

Limiting distributions of shock in critical scaling are derived.

Convergence to $F_{GOE}^2$ distribution occurs rapidly with increasing $a$.

Numerical studies illustrate the transition behavior as a function of $a$.

Abstract

We consider TASEP with two types of particles starting at every second site. Particles to the left of the origin have jump rate $1$, while particles to the right have jump rate $\alpha$. When $\alpha<1$ there is a formation of a shock where the density jumps to $(1-\alpha)/2$. For $\alpha<1$ fixed, the statistics of the associated height functions around the shock is asymptotically (as time $t\to\infty$) a maximum of two independent random variables as shown in\cite{FN14}. In this paper we consider the critical scaling when $1-\alpha=a t^{-1/3}$, where $t\gg 1$ is the observation time. In that case the decoupling does not occur anymore. We determine the limiting distributions of the shock and numerically study its convergence as a function of $a$. We see that the convergence to $F_{\rm GOE}^2$ occurs quite rapidly as $a$ increases. The critical scaling is analogue to the one used in the last passage percolation to obtain the BBP transition processes\cite{BBP06}.