Edge Universality for Nonintersecting Brownian Bridges

Abstract: In this paper we study fluctuations of extreme particles of nonintersecting Brownian bridges starting from $a_1\leq a_2\leq \cdots \leq a_n$ at time $t=0$ and ending at $b_1\leq b_2\leq \cdots\leq b_n$ at time $t=1$, where $\mu_{A_n}=(1/n)\sum_{i}\delta_{a_i}, \mu_{B_n}=(1/n)\sum_i \delta_{b_i}$ are discretization of probability measures $\mu_A, \mu_B$. Under regularity assumptions of $\mu_A, \mu_B$, we show as the number of particles $n$ goes to infinity, fluctuations of extreme particles at any time $0<t<1$, after proper rescaling, are asymptotically universal, converging to the Airy point process.