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May 2013 Crossover distributions at the edge of the rarefaction fan
Ivan Corwin, Jeremy Quastel
Ann. Probab. 41(3A): 1243-1314 (May 2013). DOI: 10.1214/11-AOP725

Abstract

We consider the weakly asymmetric limit of simple exclusion process with drift to the left, starting from step Bernoulli initial data with $\rho_{-}<\rho_{+}$ so that macroscopically one has a rarefaction fan. We study the fluctuations of the process observed along slopes in the fan, which are given by the Hopf–Cole solution of the Kardar–Parisi–Zhang (KPZ) equation, with appropriate initial data. For slopes strictly inside the fan, the initial data is a Dirac delta function and the one point distribution functions have been computed in [Comm. Pure Appl. Math. 64 (2011) 466–537] and [Nuclear Phys. B 834 (2010) 523–542]. At the edge of the rarefaction fan, the initial data is one-sided Brownian. We obtain a new family of crossover distributions giving the exact one-point distributions of this process, which converge, as $T\nearrow\infty$ to those of the Airy $\mathcal{A}_{2\to \mathrm{BM}}$ process. As an application, we prove moment and large deviation estimates for the equilibrium Hopf–Cole solution of KPZ. These bounds rely on the apparently new observation that the FKG inequality holds for the stochastic heat equation. Finally, via a Feynman–Kac path integral, the KPZ equation also governs the free energy of the continuum directed polymer, and thus our formula may also be interpreted in those terms.

Citation

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Ivan Corwin. Jeremy Quastel. "Crossover distributions at the edge of the rarefaction fan." Ann. Probab. 41 (3A) 1243 - 1314, May 2013. https://doi.org/10.1214/11-AOP725

Information

Published: May 2013
First available in Project Euclid: 29 April 2013

zbMATH: 1285.82034
MathSciNet: MR3098678
Digital Object Identifier: 10.1214/11-AOP725

Subjects:
Primary: 60H15 , 82C22

Keywords: anomalous fluctuations , Asymmetric exclusion process , Directed polymers , Kardar–Parisi–Zhang equation , Random growth , Stochastic Burgers equation , Stochastic heat equation

Rights: Copyright © 2013 Institute of Mathematical Statistics

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Vol.41 • No. 3A • May 2013
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