Extreme statistics of non-intersecting Brownian paths

TL;DR

This paper derives the joint distribution of the maximum height and its location for collections of non-intersecting Brownian paths under various boundary conditions, showing convergence to Airy process-related distributions as the number of paths grows.

Contribution

It provides explicit formulas for the joint distribution of maximum height and location for non-intersecting Brownian paths with different boundary conditions, extending previous results to large N limits.

Findings

Joint distributions converge to Airy process distributions as N→∞

Established small deviation inequalities for the argmax in Brownian bridges

Formulas expressed via Fredholm determinants of path-integral kernels

Abstract

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 [MFQR13] for the joint distribution of $\mathcal{M}={\rm max}_{x\in\mathbb{R}}\{\mathcal{A}_2(x)-x^2\}$ and $\mathcal{T}={\rm argmax}_{x\in\mathbb{R}}\{\mathcal{A}_2(x)-x^2\}$, 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 [CQR13,BCR15] 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.