Edge statistics of Dyson Brownian motion

2017/12/11 by Benjamin Landon, Horng‐Tzer Yau, Landon, Benjamin +1 · 1 citation
Mathematics · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Point processes and geometric inequalities #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1712.03881

openalex publication_date 2017/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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_* ≥ N^-2/3 and no outliers, then after times t ≫ √( 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^ε / 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].

Citations

• Random matrices: Universality of local eigenvalue statistics

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