TASEP fluctuations with soft-shock initial data

Quastel, Jeremy; Rahman, Mustazee

Abstract

We consider the totally asymmetric simple exclusion process with soft-shock initial particle density, which is a step function increasing in the direction of flow and the step size chosen small to admit KPZ scaling. The initial configuration is deterministic and the dynamics create a shock. We prove that the fluctuations of a particle at the macroscopic position of the shock converge to the maximum of two independent GOE Tracy-Widom random variables, which establishes a conjecture of Ferrari and Nejjar. Furthermore, we show the joint fluctuations of particles near the shock are determined by the maximum of two lines described in terms of these two random variables. The microscopic position of the shock is then seen to be their difference. Our proofs rely on determinantal formulae and a novel factorization of the associated kernels.

Citation

Quastel, J., & Rahman, M. (2020). TASEP fluctuations with soft-shock initial data. Annales Henri Lebesgue, 3, 999-1021. https://doi.org/10.5802/ahl.52

• Journal Article Type: Article
• Acceptance Date: Oct 14, 2019
• Online Publication Date: Aug 24, 2020
• Publication Date: 2020-08
• Deposit Date: Nov 4, 2019
• Publicly Available Date: Nov 3, 2020
• Journal: Annales Henri Lebesgue
• Electronic ISSN: 2644-9463
• Publisher: École Normale Supérieure de Rennes
• Peer Reviewed: Peer Reviewed
• Volume: 3
• Pages: 999-1021
• DOI: https://doi.org/10.5802/ahl.52
• Public URL: https://durham-repository.worktribe.com/output/1284985
• Related Public URLs: https://arxiv.org/abs/1801.06143