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arXiv · 1712.03881

Edge statistics of Dyson Brownian motion

Benjamin Landon·Horng-Tzer Yau
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Abstract

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

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Benjamin Landon, Horng-Tzer Yau. 2017-12-11. Edge statistics of Dyson Brownian motion. https://arxiv.org/abs/1712.03881

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