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One-sided reflected Brownian motions and the KPZ fixed point

Published online by Cambridge University Press:  09 December 2020

Mihai Nica
Affiliation:
Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON, Canada M5S 2E4; E-mail: mnica@math.toronto.edu
Jeremy Quastel
Affiliation:
Department of Mathematics, University of Toronto, 40 St. George Street, Toronto, ON, Canada M5S 2E4; E-mail: quastel@math.toronto.edu
Daniel Remenik
Affiliation:
Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI-CNRS 2807), Universidad de Chile, Av. Beauchef 851, Torre Norte, Piso 5, Santiago, Chile; E-mail: dremenik@dim.uchile.cl

Abstract

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We consider the system of one-sided reflected Brownian motions that is in variational duality with Brownian last passage percolation. We show that it has integrable transition probabilities, expressed in terms of Hermite polynomials and hitting times of exponential random walks, and that it converges in the 1:2:3 scaling limit to the KPZ fixed point, the scaling-invariant Markov process defined in [MQR17] and believed to govern the long-time, large-scale fluctuations for all models in the KPZ universality class. Brownian last-passage percolation was shown recently in [DOV18] to converge to the Airy sheet (or directed landscape), defined there as a strong limit of a functional of the Airy line ensemble. This establishes the variational formula for the KPZ fixed point in terms of the Airy sheet.

Type
Probability
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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