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One-sided reflected Brownian motions and the KPZ fixed point

Part of: Special processes Time-dependent statistical mechanics (dynamic and nonequilibrium)

Published online by Cambridge University Press:  09 December 2020

Mihai Nica ,
Jeremy Quastel  and
Daniel Remenik
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.

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 82C22: Interacting particle systems
Type
Probability
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e63
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

References

Anderson, G.W., Guionnet, A. and Zeitouni, O.. An introduction to random matrices. Cambridge Studies in Advanced Mathematics, vol. 118. Cambridge University Press, Cambridge, 2010, xiv+492.Google Scholar
Assiotis, T., O’Connell, N. and Warren, J.. Interlacing diffusions. In: Séminaire de Probabilités L. Ed. by Donati-Martin, C., Lejay, A. and Rouault, A.. Cham: Springer International Publishing, 2019, pp. 301–380.Google Scholar
Baryshnikov, Y.. GUEs and queues. Probab. Theory Related Fields 119.2 (2001), pp. 256–274.Google Scholar
Borodin, A., Ferrari, P.L., Prähofer, M. and Sasamoto, T.. Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129.5-6 (2007), pp. 1055–1080.Google Scholar
Corwin, I. and Hammond, A.. Brownian Gibbs property for Airy line ensembles. Invent. Math. 195.2 (2014), pp. 441–508.Google Scholar
Corwin, I., Quastel, J. and Remenik, D.. Renormalization fixed point of the KPZ universality class. J. Stat. Phys. 160.4 (2015), pp. 815–834.Google Scholar
Dauvergne, D., Nica, M. and Virág, B.. The scaling limit of the longest increasing subsequence', in preparation.Google Scholar
Dauvergne, D., Ortmann, J. and Virág, B., The directed landscape', 2018, arXiv:1812.00309 [math.PR].Google Scholar
Eynard, B. and Mehta, M.L.. Matrices coupled in a chain. I. Eigenvalue correlations. J. Phys. A 31.19 (1998), pp. 4449–4456.Google Scholar
Ferrari, P.L., Spohn, H. and Weiss, T.. Scaling limit for Brownian motions with one-sided collisions. Ann. Appl. Probab. 25.3 (2015), pp. 1349–1382.Google Scholar
Glynn, P.W. and Whitt, W.. Departures from many queues in series. Ann. Appl. Probab. 1.4 (1991), pp. 546–572.CrossRefGoogle Scholar
Gorin, V. and Shkolnikov, M.. Limits of multilevel TASEP and similar processes. Ann. Inst. Henri Poincaré Probab. Stat. 51.1 (2015), pp. 18–27.Google Scholar
Gravner, J., Tracy, C. A. and Widom, H.. Limit theorems for height fluctuations in a class of discrete space and time growth models. J. Statist. Phys. 102.5-6 (2001), pp. 1085–1132.Google Scholar
Hammond, A.. A patchwork quilt sewn from Brownian fabric: regularity of polymer weight profiles in Brownian last passage percolation. Forum Math. Pi 7 (2019), e2, 69.Google Scholar
Johansson, K.. Two time distribution in Brownian directed percolation. Comm. Math. Phys. 351.2 (2017), pp. 441–492.CrossRefGoogle Scholar
Matetski, K., Quastel, J. and Remenik, D.. The KPZ fixed point', preprint, 2020, arXiv:1701.00018 [math.PR].Google Scholar
Matetski, K., Quastel, J. and Remenik, D., TASEP and generalizations: method for exact solution', in preparation.Google Scholar
Moreno Flores, G., Quastel, J. and Remenik, D., Intermediate disorder limits for directed polymers with boundary conditions', in preparation.Google Scholar
O’Connell, N.. A path-transformation for random walks and the Robinson-Schensted correspondence. Trans. Amer. Math. Soc. 355.9 (2003), pp. 3669–3697.Google Scholar
O’Connell, N. and Yor, M.. Brownian analogues of Burke’s theorem. Stochastic Process. Appl. 96.2 (2001), pp. 285–304.Google Scholar
Quastel, J.. The Kardar-Parisi-Zhang equation. In: Current developments in mathematics, 2011. Int. Press, Somerville, MA, 2011.Google Scholar
Quastel, J. and Remenik, D.. Airy processes and variational problems. In: Topics in Percolative and Disordered Systems. Eds. Ramírez, A., Arous, G. Ben, Ferrari, P.A., Newman, C., Sidoravicius, V. and Vares, M.E.. Springer Proceedings in Mathematics & Statistics, vol. 69. Springer. 2014, pp. 121–171.Google Scholar
Sarkar, S. and Virág, B.. Brownian absolute continuity of the KPZ fixed point with arbitrary initial condition', 2020. arXiv:2002.08496 [math.PR].Google Scholar
Sasamoto, T.. Spatial correlations of the 1D KPZ surface on a flat substrate. Journal of Physics A: Mathematical and General 38.33 (2005), p. L549CrossRefGoogle Scholar
Schütz, G. M.. Exact solution of the master equation for the asymmetric exclusion process. J. Statist. Phys. 88.1-2 (1997), pp. 427–445.CrossRefGoogle Scholar
Seppäläinen, T.. A scaling limit for queues in series. Ann. Appl. Probab. 7.4 (1997), pp. 855–872.Google Scholar
Simon, B.. Trace ideals and their applications. Second. Vol. 120. Mathematical Surveys and Monographs. American Mathematical Society, 2005, pp. viii+150.Google Scholar
Warren, J.. Dyson’s Brownian motions, intertwining and interlacing. Electron. J. Probab. 12 (2007), no. 19, 573–590.CrossRefGoogle Scholar
Weiss, T., Ferrari, P. and Spohn, H.. Reflected Brownian motions in the KPZ universality class. SpringerBriefs in Mathematical Physics, vol. 18. Springer, Cham, 2017, pp. vii+118.Google Scholar