Estimates for moments of random matrices with Gaussian elements

Abstract

We describe an elementary method to get non-asymptotic estimates for the moments of Hermitian random matrices whose elements are Gaussian independent random variables. We derive a system of recurrence relations for the moments and the covariance terms and develop a triangular scheme to prove the recurrence estimales. The estimates we obtain are asymptotically exact in the sense that they give exact expressions for the first terms of 1/N-expansions of the moments and covariance terms.

As the basic example, we consider the Gaussian Unitary Ensemble of random matrices (GUE). Immediate applications include the Gaussian Orthogonal Ensemble and the ensemble of Gaussian anti-symmetric Hermitian matrices. Finally we apply our method to the ensemble of N×N Gaussian Hermitian random matrices H ^(N,b) whose elements are zero outside the band of width b. The other elements are taken from GUE; the matrix obtained is renormalized by b ^−1/2. We derive estimates for the moments of H ^(N,b) and prove that the spectral norm ⋎H ^(N,b)⋎ remains bounded in the limit N, b→∞ when (log N)^3/2/b→0.