Crossover distributions at the edge of the rarefaction fan

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.

Published in at http://dx.doi.org/10.1214/11-AOP725 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org). arXiv admin note: text overlap with arXiv:1003.0443

References in corpus (11)

• Probability Distribution of the Free Energy of the Continuum Directed Random Polymer in 1+1 dimensions
• The one-dimensional KPZ equation: an exact solution and its universality
• Asymptotics in ASEP with Step Initial Condition
• Exact height distributions for the KPZ equation with narrow wedge initial condition
• The Intermediate Disorder Regime for Directed Polymers in Dimension 1+1
• Fluctuations of the one-dimensional polynuclear growth model with external sources
• The crossover regime for the weakly asymmetric simple exclusion process
• Universality of slow decorrelation in KPZ growth
• Universal and non-universal tails of distribution functions in the directed polymer and KPZ problems
• Limit processes for TASEP with shocks and rarefaction fans
• Formulas for ASEP with Two-Sided Bernoulli Initial Condition

Cited by in corpus (11)

• Log-Gamma polymer free energy fluctuations via a Fredholm determinant identity
• Current Fluctuations of the Stationary ASEP and Six-Vertex Model
• Spectral theory for the q-Boson particle system
• Lower tail of the KPZ equation
• A Pfaffian representation for flat ASEP
• Exact formulas for random growth with half-flat initial data
• Tilted elastic lines with columnar and point disorder, non-Hermitian quantum mechanics and spiked random matrices: pinning and localization
• The KPZ equation in a half space with flat initial condition and the unbinding of a directed polymer from an attractive wall
• Local Brownian property of the narrow wedge solution of the KPZ equation
• Crossover between various initial conditions in KPZ growth: flat to stationary
• Universality for multiplicative statistics of Hermitian random matrices and the integro-differential Painlevé II equation