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Crossover distributions at the edge of the rarefaction fan. (English) Zbl 1285.82034

Authors’ abstract: We consider the weakly asymmetric limit of simple exclusion processes 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 [G. Amir et al., Commun. Pure Appl. Math. 64, No. 4, 466–537 (2011; Zbl 1222.82070)] and [T. Sasamoto and H. Spohn, Nucl. Phys., B 834, No. 3, 523–542 (2010; Zbl 1204.35137)]. 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 \(\mathcal A_{2\to BM}\) 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.

MSC:

82C22 Interacting particle systems in time-dependent statistical mechanics
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
82D60 Statistical mechanics of polymers
35Q82 PDEs in connection with statistical mechanics
35R60 PDEs with randomness, stochastic partial differential equations
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