How flat is flat in random interface growth?
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- by Jeremy Quastel and Daniel Remenik PDF
- Trans. Amer. Math. Soc. 371 (2019), 6047-6085 Request permission
Abstract:
Domains of attraction are identified for the universality classes of one-point asymptotic fluctuations for the Kardar-Parisi-Zhang (KPZ) equation with general initial data. The criterion is based on a large deviation rate function for the rescaled initial data, which arises naturally from the Hopf-Cole transformation. This allows us, in particular, to distinguish the domains of attraction of curved, flat, and Brownian initial data and to identify the boundary between the curved and flat domains of attraction, which turns out to correspond to square root initial data.
The distribution of the asymptotic one-point fluctuations is characterized by means of a variational formula written in terms of certain limiting processes (arising as subsequential limits of the spatial fluctuations of the KPZ equation with narrow wedge initial data, as shown in Probab. Theory Related Fields 166 (2016), pp. 67–185) which are widely believed to coincide with the Airy$_2$ process. In order to identify these distributions for general initial data, we extend earlier results on continuum statistics of the Airy$_2$ process to probabilities involving the process on the entire line. In particular, this allows us to write an explicit Fredholm determinant formula for the case of square root initial data.
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Additional Information
- Jeremy Quastel
- Affiliation: Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, Ontario, Canada M5S 2E4
- MR Author ID: 322635
- Email: quastel@math.toronto.edu
- Daniel Remenik
- Affiliation: Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de Chile, Av. Beauchef 851, Torre Norte, Piso 5, Santiago, Chile
- MR Author ID: 798724
- Email: dremenik@dim.uchile.cl
- Received by editor(s): August 26, 2016
- Received by editor(s) in revised form: May 30, 2017
- Published electronically: January 24, 2019
- Additional Notes: The first author gratefully acknowledges financial support from the Natural Sciences and Engineering Research Council of Canada, the I. W. Killam Foundation, and the Institute for Advanced Study.
The second author was partially supported by Fondecyt Grant 1160174, by Conicyt Basal-CMM, and by Programa Iniciativa Científica Milenio grant number NC130062 through Nucleus Millenium Stochastic Models of Complex and Disordered Systems. He also thanks the Institute for Advanced Study for its hospitality during a visit at which this project got started. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 6047-6085
- MSC (2010): Primary 60K35, 82C21
- DOI: https://doi.org/10.1090/tran/7338
- MathSciNet review: 3937318