Mesoscopic Linear Statistics of Wigner Matrices of Mixed Symmetry Class

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review

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Original languageEnglish
Pages (from-to)932-959
Journal / PublicationJournal of Statistical Physics
Issue number5
Online published19 Mar 2019
Publication statusPublished - 15 Jun 2019
Externally publishedYes


We prove a central limit theorem for the mesoscopic linear statistics of N × N Wigner matrices H satisfying E|Hij|= 1/N and EH2ij = σ/N, where σ ∈ [−1,1]. We show that on all mesoscopic scales η (1/N η ≪ 1), the linear statistics of H have a sharp transition at 1 − σ ∼ η. As an application, we identify the mesoscopic linear statistics of Dyson’s Brownian motion Ht started from a real symmetric Wigner matrix H0 at any nonnegative time ∈ [0,∞]. In particular, we obtain the transition from the central limit theorem for GOE to the one for GUE at timeη.

Research Area(s)

  • Random matrix, Wigner matrix, Linear statistics, CLT

Citation Format(s)