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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

How flat is flat in random interface growth?


Authors: Jeremy Quastel and Daniel Remenik
Journal: Trans. Amer. Math. Soc. 371 (2019), 6047-6085
MSC (2010): Primary 60K35, 82C21
DOI: https://doi.org/10.1090/tran/7338
Published electronically: January 24, 2019
MathSciNet review: 3937318
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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
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
Email: dremenik@dim.uchile.cl

DOI: https://doi.org/10.1090/tran/7338
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.
Article copyright: © Copyright 2019 American Mathematical Society