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2022 Exact lower-tail large deviations of the KPZ equation
Li-Cheng Tsai
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Duke Math. J. Advance Publication 1-44 (2022). DOI: 10.1215/00127094-2022-0008

Abstract

Consider the Hopf–Cole solution h(t,x) of the Kandar–Parisi–Zhang (KPZ) equation with the narrow wedge initial condition. Regarding t as a scaling parameter, we provide the first rigorous proof of the large deviation principle (LDP) for the lower tail of h(2t,0)+t12, with speed t2 and an explicit rate function Φ(z). This result confirms existing physics predictions made by Corwin (2011); Sasorov, Meerson, and Prolhac (2017); and Krajenbrink, Le Doussal, and Prolhac (2018). Our analysis utilizes a formula from Borodin and Gorin (2016) to convert the LDP for the KPZ equation to calculating an exponential moment of the Airy point process (PP). To estimate this exponential moment, we invoke the stochastic Airy operator (SAO) and use the Riccati transform, comparison techniques, and certain variational characterizations of the relevant functional.

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Li-Cheng Tsai. "Exact lower-tail large deviations of the KPZ equation." Duke Math. J. Advance Publication 1 - 44, 2022. https://doi.org/10.1215/00127094-2022-0008

Information

Received: 25 February 2020; Revised: 17 May 2021; Published: 2022
First available in Project Euclid: 10 May 2022

Digital Object Identifier: 10.1215/00127094-2022-0008

Subjects:
Primary: 60F10
Secondary: 60H25

Keywords: Airy point process , Kardar–Parisi–Zhang equation , large deviations , Random operators , Stochastic Airy operator

Rights: Copyright © 2022 Duke University Press

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