Local law and mesoscopic fluctuations of Dyson Brownian motion for general $\beta$ and potential

Abstract

We study Dyson Brownian motion with general potential $V$ and for general $\beta \ge 1$. For short times $t=o(1)$ and under suitable conditions on $V$ we obtain a local law and corresponding rigidity estimates on the particle locations; that is, with overwhelming probability, the particles are close to their classical locations with an almost-optimal error estimate. Under the condition that the density of states of the initial data is bounded below and above down to the scale ${\eta }_{\ast }\ll t\ll 1$, we prove a mesoscopic central limit theorem for linear statistics at all scales $N^{-1}\ll \eta \ll t$

References in corpus (2)

• Local semicircle law and complete delocalization for Wigner random matrices
• Generalized Dyson Brownian motion, McKean-Vlasov equation and eigenvalues of random matrices

Cited by in corpus (7)

• Non-Ergodic Delocalization in the Rosenzweig-Porter Model
• Edge statistics of Dyson Brownian motion
• The Dyson and Coulomb games
• Local spectral statistics of the addition of random matrices
• Universality of the least singular value for the sum of random matrices
• Tridiagonal Models for Dyson Brownian Motion
• $\beta$-Nonintersecting Poisson Random Walks: Law of Large Numbers and Central Limit Theorems