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Limit Processes for TASEP with Shocks and Rarefaction Fans

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Abstract

We consider the totally asymmetric simple exclusion process (TASEP) with two-sided Bernoulli initial condition, i.e., with left density ρ − and right density ρ +. We study 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 ρ ±. We characterize the large time scaling limit of the multipoint fluctuations as a function of the densities ρ ± 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.

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012, USA

    Ivan Corwin

  2. Institute for Applied Mathematics, University of Bonn, Endenicher Allee 60, 53115, Bonn, Germany

    Patrik L. Ferrari

  3. Institut Fourier, 100 Rue des maths, 38402, Saint Martin d’Heres, France

    Sandrine Péché

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  1. Ivan Corwin
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  2. Patrik L. Ferrari
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  3. Sandrine Péché
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Correspondence to Patrik L. Ferrari.

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Corwin, I., Ferrari, P.L. & Péché, S. Limit Processes for TASEP with Shocks and Rarefaction Fans. J Stat Phys 140, 232–267 (2010). https://doi.org/10.1007/s10955-010-9995-7

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