November 2022 Airy process with wanderers, KPZ fluctuations, and a deformation of the Tracy–Widom GOE distribution
Karl Liechty, Gia Bao Nguyen, Daniel Remenik
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 58(4): 2250-2283 (November 2022). DOI: 10.1214/21-AIHP1229


We study the distribution of the supremum of the Airy process with m wanderers minus a parabola, or equivalently the limit of the rescaled maximal height of a system of N non-intersecting Brownian bridges as N, where the first Nm paths start and end at the origin and the remaining m go between arbitrary positions. The distribution provides a 2m-parameter deformation of the Tracy–Widom GOE distribution, which is recovered in the limit corresponding to all Brownian paths starting and ending at the origin.

We provide several descriptions of this distribution function: (i) A Fredholm determinant formula; (ii) A formula in terms of Painlevé II functions; (iii) A representation as a marginal of the KPZ fixed point with initial data given as the top path in a stationary system of reflected Brownian motions with drift; (iv) A characterization as the solution of a version of the Bloemendal–Virag PDE (Probab. Theory Related Fields 156 (2013) 795–825; Ann. Probab. 44 (2016) 2726–2769) for spiked Tracy–Widom distributions; (v) A representation as a solution of the KdV equation. We also discuss connections with a model of last passage percolation with boundary sources.

Nous étudions la loi du supremum du processus d’Airy à m promeneurs, moins une parabole, ou de manière équivalente, la limite de la hauteur maximale rééchelonnée d’un système de N ponts browniens non intersectants, lorsque N, où les Nm premiers chemins commencent et terminent à l’origine et les m restants lient des positions arbitraires. Cette loi fournit une déformation à 2m paramètres de la loi GOE de Tracy–Widom, qui est retrouvée comme limite lorsque tous les chemins browniens qui commencent et terminent à l’origine. Nous fournissons plusieurs descriptions de cette loi : (i) une formule utilisant un déterminant de Fredholm ; (ii) une formule en termes de fonctions de Painlevé II ; (iii) une représentation en tant que marginal du point fixe KPZ, avec des données initiales données par le chemin supérieur d’un système stationnaire de mouvements browniens avec dérive réfléchis ; (iv) une caractérisation comme solution d’une version de l’EDP de Bloemendal–Virag (Probab. Theory Related Fields 156 (2013) 795–825 ; Ann. Probab. 44 (2016) 2726–2769) pour des lois de Tracy–Widom avec pointes ; (v) une représentation en tant que solution de l’équation KdV. Nous discutons aussi des liens avec un modèle de percolation de dernier passage avec source à la frontière.

Funding Statement

KL was supported by a Simons Foundation Collaboration Grant #357872 as well as a Faculty Summer Research Grant from the College of Science and Health at DePaul University.
GBN was supported by the Swedish Research Council Grant 67465 VR20BN.
DR was supported by Centro de Modelamiento Matemático (CMM) Basal Funds AFB170001, ACE210010 and FB210005 from ANID-Chile, by Programa Iniciativa Científica Milenio grant number NC120062 through Nucleus Millenium Stochastic Models of Complex and Disordered Systems, and by Fondecyt Grant 1201914.


The authors thank Mauricio Duarte for background and references on RBMs, Joaquín Fontbona for helping us with the abstract argument which proves the convergence to a stationary process for the system of RBMs with drift, and Dan Betea for pointing out to us that Corollary 1 can be understood in terms of a symmetry of LPP with boundary sources.

Throughout this project we used Folkmar Bornemann’s MATLAB package for numerical computation of Fredholm determinants [14] and the Mathematica package RHPackage by Sheehan Olver for numerical solutions to Riemann–Hilbert problems [60] to compare the different formulas from Theorems 1 and 2 and Conjecture 1.


Download Citation

Karl Liechty. Gia Bao Nguyen. Daniel Remenik. "Airy process with wanderers, KPZ fluctuations, and a deformation of the Tracy–Widom GOE distribution." Ann. Inst. H. Poincaré Probab. Statist. 58 (4) 2250 - 2283, November 2022.


Received: 24 October 2020; Revised: 4 September 2021; Accepted: 9 November 2021; Published: November 2022
First available in Project Euclid: 6 October 2022

MathSciNet: MR4492978
zbMATH: 1498.60037
Digital Object Identifier: 10.1214/21-AIHP1229

Primary: 60B20 , 60J65 , 60K35

Keywords: Airy processes , KPZ fixed point , Non-intersecting Brownian motions , Painlevé II , random matrices

Rights: Copyright © 2022 Association des Publications de l’Institut Henri Poincaré


This article is only available to subscribers.
It is not available for individual sale.

Vol.58 • No. 4 • November 2022
Back to Top