Limit processes for TASEP with shocks and rarefaction fans

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.

46 pages, 3 figures, LaTeX; Extended explanations in the first two sections

References in corpus (15)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Asymptotics in ASEP with Step Initial Condition
• Fluctuation properties of the TASEP with periodic initial configuration
• A Fredholm Determinant Representation in ASEP
• Transition between Airy_1 and Airy_2 processes and TASEP fluctuations
• Fluctuations of the one-dimensional polynuclear growth model with external sources
• Airy kernel with two sets of parameters in directed percolation and random matrix theory
• Limit process of stationary TASEP near the characteristic line
• PDEs for the joint distributions of the Dyson, Airy and Sine processes
• Dynamics of a tagged particle in the asymmetric exclusion process with the step initial condition
• Slow decorrelations in KPZ growth
• The motion of a second class particle for the tasep starting from a decreasing shock profile
• Total Current Fluctuations in ASEP
• On ASEP with Step Bernoulli Initial Condition
• $t^{1/3}$ Superdiffusivity of Finite-Range Asymmetric Exclusion Processes on $\mathbb{Z}$

Cited by in corpus (21)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• Airy processes and variational problems
• Universality of slow decorrelation in KPZ growth
• Facilitated Asymmetric Exclusion
• On Time Correlations for KPZ Growth in One Dimension
• Time-time covariance for last passage percolation with generic initial profile
• Transversal fluctuations of the ASEP, stochastic six vertex model, and Hall-Littlewood Gibbsian line ensembles
• Replica approach to the KPZ equation with half Brownian motion initial condition
• Crossover distributions at the edge of the rarefaction fan
• The one-dimensional Kardar-Parisi-Zhang and Kuramoto-Sivashinsky universality class: limit distributions
• KP governs random growth off a one dimensional substrate
• Interface dynamics in the two-dimensional quantum Ising model
• Supremum of the Airy2 process minus a parabola on a half line
• TASEP and generalizations: Method for exact solution
• The KPZ equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall
• Directed last passage percolation with discontinuous weights
• Shock fluctuations in flat TASEP under critical scaling
• Finite GUE distribution with cut-off at a shock
• Airy process with wanderers, KPZ fluctuations, and a deformation of the Tracy--Widom GOE distribution
• TASEP fluctuations with soft-shock initial data
• TASEP with a general initial condition and a deterministically moving wall