Convergence of the empirical spectral measure of unitary Brownian motion

2017/05/08 by Elizabeth Meckes, Meckes, Elizabeth, Tai Melcher +1
Mathematics · #58J65 #60B20 #Advanced Algebra and Geometry #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.1705.03057

openalex publication_date 2017/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \U^N_t\_{t≥ 0} be a standard Brownian motion on \mathbbU(N). For fixed N∈ℕ and t>0, we give explicit bounds on the L₁-Wasserstein distance of the empirical spectral measure of U^N_t to both the ensemble-averaged spectral measure and to the large-N limiting measure identified by Biane. We are then able to use these bounds to control the rate of convergence of paths of the measures on compact time intervals. The proofs use tools developed by the first author to study convergence rates of the classical random matrix ensembles, as well as recent estimates for the convergence of the moments of the ensemble-average spectral distribution.

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