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 η_* ≥ N-2/3 and no outliers, then after times t √ η_*, the statistics at the spectral edge agree with the GOE/GUE. In particular we obtain the optimal time to equilibrium at the edge t = Nε / N1/3 for sufficiently regular initial data. Our methods rely on eigenvalue rigidity results similar to those appearing in [Lee-Schnelli], the coupling idea of [Bourgade-Erdős-Yau-Yin] and the energy estimate of [Bourgade-Erdős-Yau].
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Landon et al. (2017) studied this question.
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