Moderate deviations for spectral measures of random matrix ensembles

Abstract

In this paper we consider the (weighted) spectral measure ${\mu }_n$ of a $n\times n$ random matrix, distributed according to a classical Gaussian, Laguerre or Jacobi ensemble, and show a moderate deviation principle for the standardised signed measure $\sqrt{n/a_n}({\mu }_n-\sigma )$. The centering measure $\sigma$ is the weak limit of the empirical eigenvalue distribution and the rate function is given in terms of the $L^2$-norm of the density with respect to $\sigma$. The proof involves the tridiagonal representations of the ensembles which provide us with a sequence of independent random variables and a link to orthogonal polynomials.

20 pages

References in corpus (4)

• Central limit theorem for linear eigenvalue statistics of random matrices with independent entries
• Canonical moments and random spectral measures
• Large deviations for functions of two random projection matrices
• Moderate deviations for the eigenvalue counting function of Wigner matrices

Cited by in corpus (1)

• On spectral measures of random Jacobi matrices