Restricted maximum of non-intersecting Brownian bridges

¹ Departamento de Ingeniería Matemática, Universidad de Chile, Santiago, Chile
² Université de Lorraine, CNRS, Inria, IECL, UMR 7502, 54000 Nancy, France

^* Corresponding author: yamit@dim.uchile.cl

Received: 24 October 2023
Accepted: 1 April 2024

Abstract

Consider a system of N non-intersecting Brownian bridges in [0,1], and let ℳ_N(p) be the maximal height attained by the top path in the interval [0, p], p ∈ [0, 1]. It is known that, under a suitable rescaling, the distribution of ℳ_N(p) converges, as N → ∞, to a one-parameter family of distributions interpolating between the Tracy-Widom distributions for the Gaussian Orthogonal and Unitary Ensembles (corresponding, respectively, to p → 1 and p → 0). It is also known that, for fixed N, ℳ_N(1) is distributed as the top eigenvalue of a random matrix drawn from the Laguerre Orthogonal Ensemble. Here we show a version of these results for ℳ_N(p) for fixed N, showing that ℳ_N(p) / √p converges in distribution, as p → 0, to the rightmost charge in a generalized Laguerre Unitary Ensemble, which coincides with the top eigenvalue of a random matrix drawn from the Antisymmetric Gaussian Ensemble.