2021/03/31 by Paul Bourgade, Krishnan Mody, Michel Pain
Mathematics · Physics and Astronomy · #Central limit theorem #Combinatorics #Covariance #Eigenvalues and eigenvectors #Gaussian #Law of large numbers #Logarithm #Mathematical analysis #Mathematical physics #Mathematics #Physics #Pure mathematics #Quantum mechanics #Random Matrices and Applications #Random matrix #Random variable #Statistical physics #Statistics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Universality (dynamical systems) #math-ph #math.MP #math.PR
paper · pdf · doi:10.1007/s00220-022-04311-2
47 pages, to appear in Comm. Math. Phys
arxiv created 2022/01/17 · openalex publication_date 2022/01/29 · arxiv updated 2022/03/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
In the setting of generic β-ensembles, we use the loop equation hierarchy to prove a local law with optimal error up to a constant, valid on any scale including microscopic. This local law has the following consequences. (i) The optimal rigidity scale of the ordered particles is of order (log N)/N in the bulk of the spectrum. (ii) Fluctuations of the particles satisfy a central limit theorem with covariance corresponding to a logarithmically correlated field; in particular each particle in the bulk fluctuates on scale √(log N)/N. (iii) The logarithm of the electric potential also satisfies a logarithmically correlated central limit theorem. Contrary to much progress on random matrix universality, these results do not proceed by comparison. Indeed, they are new for the Gaussian β-ensembles. By comparison techniques, (ii) and (iii) also hold for Wigner matrices.