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 (10)

• Level-Spacing Distributions and the Airy Kernel
• Random matrices: Universality of local eigenvalue statistics up to the edge
• Edge Universality for Deformed Wigner Matrices
• Bulk universality for deformed Wigner matrices
• Quadratic vector equations on complex upper half-plane
• Bulk universality of sparse random matrices
• Local spectral statistics of Gaussian matrices with correlated entries
• Spectral Statistics of Sparse Random Graphs with a General Degree Distribution
• Local law and mesoscopic fluctuations of Dyson Brownian motion for general $\beta$ and potential
• Local spectral statistics of the addition of random matrices

Cited by in corpus (8)

• Edge Universality for non-Hermitian Random Matrices
• Edge rigidity and universality of random regular graphs of intermediate degree
• Cusp Universality for Random Matrices I: Local Law and the Complex Hermitian Case
• Transition from Tracy-Widom to Gaussian fluctuations of extremal eigenvalues of sparse Erdős-Rényi graphs
• Quenched universality for deformed Wigner matrices
• Edge Universality for Nonintersecting Brownian Bridges
• The Edge Universality of Correlated Matrices
• Edge scaling limit of Dyson Brownian motion at equilibrium for general $\beta \ge 1$