Open Access
2017 Extreme statistics of non-intersecting Brownian paths
Gia Bao Nguyen, Daniel Remenik
Electron. J. Probab. 22: 1-40 (2017). DOI: 10.1214/17-EJP119


We consider finite collections of $N$ non-intersecting Brownian paths on the line and on the half-line with both absorbing and reflecting boundary conditions (corresponding to Brownian excursions and reflected Brownian motions) and compute in each case the joint distribution of the maximal height of the top path and the location at which this maximum is attained. The resulting formulas are analogous to the ones obtained in [28] for the joint distribution of $\mathcal{M} =\max _{x\in \mathbb{R} }\!\big \{\mathcal{A} _2(x)-x^2\}$ and $\mathcal{T} =\operatorname{argmax} _{x\in \mathbb{R} }\!\big \{\mathcal{A} _2(x)-x^2\}$, where $\mathcal{A} _2$ is the Airy$_2$ process, and we use them to show that in the three cases the joint distribution converges, as $N\to \infty $, to the joint distribution of $\mathcal{M} $ and $\mathcal{T} $. In the case of non-intersecting Brownian bridges on the line, we also establish small deviation inequalities for the argmax which match the tail behavior of $\mathcal{T} $. Our proofs are based on the method introduced in [9, 6] for obtaining formulas for the probability that the top line of these line ensembles stays below a given curve, which are given in terms of the Fredholm determinant of certain “path-integral” kernels.


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Gia Bao Nguyen. Daniel Remenik. "Extreme statistics of non-intersecting Brownian paths." Electron. J. Probab. 22 1 - 40, 2017.


Received: 10 January 2017; Accepted: 22 October 2017; Published: 2017
First available in Project Euclid: 27 November 2017

zbMATH: 06827079
MathSciNet: MR3733660
Digital Object Identifier: 10.1214/17-EJP119

Primary: 60B20 , 60J65 , 82C23

Keywords: Airy process , KPZ universality class , Non-intersecting Brownian motions , polymer endpoint distribution

Vol.22 • 2017
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