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.
[BB19] Color-position symmetry in interacting particle systems (2019) (https://arxiv.org/abs/1905.04692)
[BG16] Lectures on integrable probability, Probability and Statistical Physics in St. Petersburg (Sidoravicious, V.; Smirnov, V., eds.) (Proceedings of Symposia in Pure Mathematics) Volume 91, American Mathematical Society, 2016, pp. 155-214 | MR | Zbl
[Fer94] Shocks in one-dimensional processes with drift, Probability and Phase Transition (Grimmett, Geoffrey, ed.) (NATO ASI Series. Series C. Mathematical and Physical Sciences) Volume 420, Kluwer Academic Publishers, 1994, pp. 35-48 | DOI | MR | Zbl
[FFV00] The asymmetric simple exclusion process with multiple shocks, Ann. Inst. Henri Poincaré, Probab. Stat., Volume 36 (2000) no. 2, pp. 109-126 | DOI
[Nej19] GUE GUE limit law at hard shocks in ASEP (2019) (https://arxiv.org/abs/1906.07711)
[OODL + 17] NIST. Digital Library of Mathematical Functions, 2017 (http://dlmf.nist.gov/,Release 1.0.16)
[Qua13] Introduction to KPZ, Current Developments in Mathematics, Volume 2011, International Press, 2013, pp. 125-194 | Zbl