Edge statistics of Dyson Brownian motion

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 ${\eta }_{\ast }\ge N^{-2/3}$ 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 $t=N^{\epsilon }/N^{1/3}$ 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].

References in corpus (1)

• Local spectral statistics of the addition of random matrices

Cited by in corpus (3)

• Transition from Tracy-Widom to Gaussian fluctuations of extremal eigenvalues of sparse Erdős-Rényi graphs
• The Edge Universality of Correlated Matrices
• Edge scaling limit of Dyson Brownian motion at equilibrium for general $\beta \ge 1$