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تسجيل مستخدم جديدEdge statistics of Dyson Brownian motion
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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 $eta_* geq N^{-2/3}$ and no outliers, then after times $t gg sqrt{ eta_*}$, the statistics at the spectral edge agree with the GOE/GUE. In particular we obtain the optimal time to equilibrium at the edge $t = N^{varepsilon} / N^{1/3}$ for sufficiently regular initial data. Our methods rely on eigenvalue rigidity results similar to those appearing in [Lee-Schnelli], the coupling idea of [Bourgade-ErdH{o}s-Yau-Yin] and the energy estimate of [Bourgade-ErdH{o}s-Yau].
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