Coulomb-Gas Electrostatics Controls Large Fluctuations of the Kardar-Parisi-Zhang Equation

Ivan Corwin, Promit Ghosal, Alexandre Krajenbrink, Pierre Le Doussal, and Li-Cheng Tsai
Phys. Rev. Lett. 121, 060201 – Published 8 August 2018
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Abstract

We establish a large deviation principle for the Kardar-Parisi-Zhang (KPZ) equation, providing precise control over the left tail of the height distribution for narrow wedge initial condition. Our analysis exploits an exact connection between the KPZ one-point distribution and the Airy point process—an infinite particle Coulomb gas that arises at the spectral edge in random matrix theory. We develop the large deviation principle for the Airy point process and use it to compute, in a straightforward and assumption-free manner, the KPZ large deviation rate function in terms of an electrostatic problem (whose solution we evaluate). This method also applies to the half-space KPZ equation, showing that its rate function is half of the full-space rate function. In addition to these long-time estimates, we provide rigorous proof of finite-time tail bounds on the KPZ distribution, which demonstrate a crossover between exponential decay with exponent 3 (in the shallow left tail) to exponent 5/2 (in the deep left tail). The full-space KPZ rate function agrees with the one computed in Sasorov et al. [J. Stat. Mech. (2017) 063203] via a WKB approximation analysis of a nonlocal, nonlinear integrodifferential equation generalizing Painlevé II which Amir et al. [Commun. Pure Appl. Math. 64, 466 (2011)] related to the KPZ one-point distribution.

  • Figure
  • Received 27 March 2018

DOI:https://doi.org/10.1103/PhysRevLett.121.060201

© 2018 American Physical Society

Physics Subject Headings (PhySH)

Statistical Physics & Thermodynamics

Authors & Affiliations

Ivan Corwin1, Promit Ghosal2, Alexandre Krajenbrink3, Pierre Le Doussal3, and Li-Cheng Tsai1

  • 1Columbia University, Department of Mathematics 2990 Broadway, New York, New York 10027, USA
  • 2Columbia University, Department of Statistics 1255 Amsterdam, New York, New York 10027, USA
  • 3CNRS—Laboratoire de Physique Théorique de l’Ecole Normale Supérieure, 24 rue Lhomond, 75231 Paris Cedex, France

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Issue

Vol. 121, Iss. 6 — 10 August 2018

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