Moderate deviations for the spectral measure of certain random matrices. (Q1413102)

The authors derive a moderate deviation principle for matrices of the form \(X_N=D_N+W_N\) where \(W_N\) are Wigner matrices, that is, a symmetric or Hermitian matrix with real (respectively complex) i.i.d Gaussian entries of covariance \(N^{-1}\) above the diagonal, and \(D_N\) is a sequence of diagonal matrices with elements \(d_i^N\) and spectral measure \(N^{-1}\sum_{i=1}^N\delta_{d_i^N}\) that converges in a strong sense to a limit \(\mu_D\). Denoting by \(\hat\mu^N_{X_N}\) the spectral measure of \(X_N\), large deviation (in the scale \(N^2\)) and CLT are obtained in the literature by a dynamical approach based that \(W_N\) can be constructed as a Hermitian or symmetric Brownian motion in time one. The authors extend this analysis to study the moderate deviation for \(\hat\mu^N_{X_N}\).