Mesoscopic Linear Statistics of Wigner Matrices of Mixed Symmetry Class

We prove a central limit theorem for the mesoscopic linear statistics of N × N Wigner matrices H satisfying E|H_ij|² = 1/N and EH²_ij = 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 H_t started from a real symmetric Wigner matrix H₀ at any nonnegative time t ∈ [0,∞]. In particular, we obtain the transition from the central limit theorem for GOE to the one for GUE at time t ∼ η.