Finite GUE distribution with cut-off at a shock

TL;DR

This paper studies the fluctuations in a particle system with shocks, revealing a finite GUE distribution with a cut-off, using a novel approach that avoids traditional last passage percolation techniques.

Contribution

It introduces a new probabilistic method to analyze space-time correlations in asymmetric exclusion processes with shocks, bypassing the need for last passage percolation mapping.

Findings

Fluctuations are governed by a cut-off GUE distribution.

Correlation structure described without last passage percolation.

Special case links to largest eigenvalue distribution of finite GUE.

Abstract

We consider the totally asymmetric simple exclusion process with initial conditions generating a shock. The fluctuations of particle positions are asymptotically governed by the randomness around the two characteristic lines joining at the shock. Unlike in previous papers, we describe the correlation in space-time \emph{without} employing the mapping to the last passage percolation, which fails to exists already for the partially asymmetric model. We then consider a special case, where the asymptotic distribution is a cut-off of the distribution of the largest eigenvalue of a finite GUE matrix. Finally we discuss the strength of the probabilistic and physically motivated approach and compare it with the mathematical difficulties of a direct computation.