[Paper Review] Limit processes of non-equilibrium TASEP

Ivan Corwin, Patrik L. Ferrari+1 more|arXiv (Cornell University)|Feb 18, 2010

Stochastic processes and statistical mechanics7 citations

TL;DR

This paper studies the macroscopic and mesoscopic fluctuations of the totally asymmetric simple exclusion process (TASEP) with two-sided Bernoulli initial conditions, where particles are injected with left density ρ₋ and right density ρ₊. Using a slow decorrelation principle, it establishes the large-time scaling limits of height function fluctuations across different macroscopic regions, extending results from fixed time to the full space-time plane except along characteristic curves of the Burgers equation, with analogous results derived for last passage percolation and eigenvalue distributions of perturbed Wishart matrices.

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.

Motivation & Objective

• To characterize the large-time scaling limits of height function fluctuations in TASEP with two-sided Bernoulli initial conditions.
• To extend fixed-time fluctuation results to the full space-time plane, except along characteristic directions of the Burgers equation.
• To establish analogous fluctuation limits for corner growth processes with external sources and last passage percolation with two-sided boundary conditions.
• To derive corresponding results for eigenvalues of perturbed complex Wishart matrices.

Proposed method

• Analyzes the TASEP height function under two-sided Bernoulli initial conditions with densities ρ₋ and ρ₊.
• Applies the slow decorrelation principle to extend fixed-time fluctuation results to the full space-time domain.
• Uses the connection between TASEP and last passage percolation with two-sided boundary conditions to derive fluctuation limits.
• Establishes limit processes for corner growth models with external sources, equivalent to last passage times in directed percolation.
• Applies the same framework to study eigenvalue fluctuations in perturbed complex Wishart matrices.
• Relies on macroscopic limit shape analysis and region-specific fluctuation scaling, depending on the relative values of ρ₊ and ρ₋.

Experimental results

Research questions

• RQ1 How do the fluctuations of the TASEP height function scale in the large-time limit under two-sided Bernoulli initial conditions with densities ρ₋ and ρ₊?
• RQ2 What is the behavior of these fluctuations across different macroscopic regions defined by the relative values of ρ₊ and ρ₋?
• RQ3 To what extent can fixed-time fluctuation results be extended to the full space-time plane in TASEP?
• RQ4 What are the limiting processes for last passage percolation with two-sided boundary conditions and external sources?
• RQ5 How do the eigenvalue fluctuations of perturbed complex Wishart matrices relate to the TASEP and percolation models studied?

Key findings

• The large-time scaling limit of TASEP height function fluctuations depends on the macroscopic region defined by the densities ρ₊ and ρ₋, with distinct behaviors in shock and rarefaction fan regions.
• Fluctuations are characterized by universal limit processes that vary continuously across the macroscopic phase diagram.
• The slow decorrelation principle enables extension of fixed-time fluctuation results to the full space-time plane, except along characteristic curves of the Burgers equation.
• Analogous fluctuation limits are derived for corner growth processes with external sources and for last passage percolation with two-sided boundary conditions.
• The same limiting processes are shown to govern the eigenvalue fluctuations of perturbed complex Wishart matrices.
• The results establish a unified framework linking TASEP, directed percolation, and random matrix theory through universal scaling limits.