Extreme statistics of non-intersecting Brownian paths

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 $\mathcal{M}={\max }_{x\in \mathbb{R}}\{{\mathcal{A}}_2(x)-x^2\}$ and $\mathcal{T}={\mathrm{argmax}}_{x\in \mathbb{R}}\{{\mathcal{A}}_2(x)-x^2\}$, 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.

Minor corrections, improved exposition. To appear in Electron. J. Probab

References in corpus (3)

• The one-dimensional KPZ equation and its universality class
• Maximum distributions of bridges of noncolliding Brownian paths
• The hard-edge tacnode process for Brownian motion

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• KPZ fluctuations in finite volume
• Constrained non-crossing Brownian motions, fermions and the Ferrari-Spohn distribution
• A stationary model of non-intersecting directed polymers