Edge statistics of Dyson Brownian motion

We consider the edge statistics of Dyson Brownian motion with deterministic initial data. Our main result states that if the initial data has a spectral edge with rough square root behavior down to a scale ${\eta }_{\ast }\ge N^{-2\mathrm{/}3}$ and no outliers, then after times $t\gg \sqrt{{\eta }_{\ast }}$, the statistics at the spectral edge agree with the GOE/GUE. In particular we obtain the optimal time to equilibrium at the edge $t=N^{\epsilon }\mathrm{/}{N}^{1\mathrm{/}3}$ for sufficiently regular initial data. Our methods rely on eigenvalue rigidity results similar to those appearing in [Lee-Schnelli], the coupling idea of [Bourgade-Erd\H{o}s-Yau-Yin] and the energy estimate of [Bourgade-Erd\H{o}s-Yau].