Limit processes for TASEP with shocks and rarefaction fans

TL;DR

This paper analyzes the large-scale fluctuation limits of TASEP with two-sided Bernoulli initial conditions, revealing how shock and rarefaction regions influence the asymptotic behavior of the height function and related processes.

Contribution

It characterizes the fluctuation limits of TASEP with shocks and rarefaction fans, extending results via slow decorrelation, and connects these to last passage percolation and eigenvalue distributions.

Findings

Fluctuation limits depend on initial densities and macroscopic regions.

Results extend from fixed time to space-time, except along characteristic directions.

Provides limit processes for related growth models and eigenvalue distributions.

Abstract

We consider the totally asymmetric simple exclusion process (TASEP) with two-sided Bernoulli initial condition, i.e., with left density rho_- and right density rho_+. We consider the associated height function, whose discrete gradient is given by the particle occurrences. Macroscopically one has a deterministic limit shape with a shock or a rarefaction fan depending on the values of rho_{+/-}. We characterize the large time scaling limit of the fluctuations as a function of the densities rho_{+/-} and of the different macroscopic regions. Moreover, using a slow decorrelation phenomena, the results are extended from fixed time to the whole space-time, except along the some directions (the characteristic solutions of the related Burgers equation) where the problem is still open. On the way to proving the results for TASEP, we obtain the limit processes for the fluctuations in a class of corner growth processes with external sources, of equivalently for the last passage time in a directed percolation model with two-sided boundary conditions. Additionally, we provide analogous results for eigenvalues of perturbed complex Wishart (sample covariance) matrices.