Electronic Journal of Probability

Extreme statistics of non-intersecting Brownian paths

Gia Bao Nguyen and Daniel Remenik

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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.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 102, 40 pp.

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

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Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65] 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 82C23: Exactly solvable dynamic models [See also 37K60]

non-intersecting Brownian motions KPZ universality class Airy process polymer endpoint distribution

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Nguyen, Gia Bao; Remenik, Daniel. Extreme statistics of non-intersecting Brownian paths. Electron. J. Probab. 22 (2017), paper no. 102, 40 pp. doi:10.1214/17-EJP119. https://projecteuclid.org/euclid.ejp/1511773232

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