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.
On considère le processus d’exclusion simple totalement asymétrique avec une densité initiale de particules à choc modéré, ayant la forme d’une fonction en escalier qui augmente dans la direction du flux et dont le saut est choisi suffisamment petit pour satisfaire l’échelle KPZ. On suppose que la configuration initiale est déterministe et que la dynamique crée un choc.
On montre que les fluctuations d’une particule située à l’endroit macroscopique du choc convergent vers le maximum de deux variables aléatoires indépendantes de type GOE Tracy–Widom. Ceci prouve une conjecture de Ferrari et Nejjar. On montre également que les fluctuations conjointes des particules près du choc sont décrites par le maximum de deux lignes définies en termes de ces deux variables aléatoires. La position microscopique du choc apparaît alors comme leur différence.
Nos preuves sont basées sur des formules déterminantes et une nouvelle factorisation des noyaux associés.
Revised:
Accepted:
Published online:
Keywords: Burgers equation, exclusion process, KPZ universality, shock fluctuations, Tracy–Widom law
@article{AHL_2020__3__999_0, author = {Quastel, Jeremy and Rahman, Mustazee}, title = {TASEP fluctuations with soft-shock initial data}, journal = {Annales Henri Lebesgue}, pages = {999--1021}, publisher = {\'ENS Rennes}, volume = {3}, year = {2020}, doi = {10.5802/ahl.52}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/ahl.52/} }
Quastel, Jeremy; Rahman, Mustazee. TASEP fluctuations with soft-shock initial data. Annales Henri Lebesgue, Volume 3 (2020), pp. 999-1021. doi : 10.5802/ahl.52. http://archive.numdam.org/articles/10.5802/ahl.52/
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