TASEP fluctuations with soft-shock initial data

TL;DR

This paper studies the fluctuations in the TASEP with soft-shock initial data, showing they converge to the maximum of two GOE Tracy-Widom distributions, confirming a conjecture and revealing joint fluctuation behavior.

Contribution

It proves the conjecture that TASEP shock fluctuations are governed by the maximum of two GOE Tracy-Widom variables and introduces a new kernel factorization method.

Findings

Fluctuations at the shock converge to the maximum of two GOE Tracy-Widom distributions.

Joint fluctuations near the shock are determined by these two distributions.

The microscopic shock position is given by their difference.

Abstract

We consider the totally asymmetric simple exclusion process with \emph{soft-shock} initial particle density, which is a step function increasing in the direction of flow and the step size chosen small to admit KPZ scaling. The initial configuration is deterministic and the dynamics create a shock. We prove that the fluctuations of a particle at the macroscopic position of the shock converge to the maximum of two independent GOE Tracy-Widom random variables, which establishes a conjecture of Ferrari and Nejjar. Furthermore, we show the joint fluctuations of particles near the shock are determined by the maximum of two lines described in terms of these two random variables. The microscopic position of the shock is then seen to be their difference. Our proofs rely on determinantal formulae and a novel factorization of the associated kernels.