Restricted Maximum of Non-Intersecting Brownian Bridges

TL;DR

This paper investigates the maximal height of the top path in a system of non-intersecting Brownian bridges, revealing new connections to random matrix ensembles and their eigenvalue distributions.

Contribution

It extends known results by analyzing the distribution of the maximal height for fixed number of paths, linking it to the Antisymmetric Gaussian Ensemble.

Findings

As p approaches 0, the scaled maximum converges to the top eigenvalue of the Antisymmetric Gaussian Ensemble.

For fixed N, the distribution of the maximum relates to the generalized Laguerre Unitary Ensemble.

Results interpolate between Tracy-Widom distributions for different ensembles.

Abstract

Consider a system of $N$ non-intersecting Brownian bridges in $[0,1]$, and let $\mathcal M_N(p)$ be the maximal height attained by the top path in the interval $[0,p]$, $p\in[0,1]$. It is known that, under a suitable rescaling, the distribution of $\mathcal M_N(p)$ converges, as $N\to\infty$, to a one-parameter family of distributions interpolating between the Tracy-Widom distributions for the Gaussian Orthogonal and Unitary Ensembles (corresponding, respectively, to $p\to1$ and $p\to0$). It is also known that, for fixed $N$, $\mathcal M_N(1)$ 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 $\mathcal M_N(p)$ for fixed $N$, showing that $\mathcal M_N(p)/\sqrt{p}$ converges in distribution, as $p\to0$, 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.