Restricted Maximum of Non-Intersecting Brownian Bridges

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.