Perturbations of Diagonal Matrices by Band Random Matrices

We exhibit an explicit formula for the spectral density of a (large) random matrix which is a diagonal matrix whose spectral density converges, perturbated by the addition of a symmetric matrix with Gaussian entries and a given (small) limiting variance profile. 1. Perturbation of the spectral density of a large diagonal matrix In this paper, we consider the spectral measure of a random matrix Dε n defined by D ε n = Dn + √ ε nXn, for Dn a deterministic diagonal matrix whose spectral measure converges and Xn an Hermitian or real symmetric matrix whose entries are Gaussian independent variables, with a limiting variance profile (such matrices are called band matrices). We give a first order Taylor expansion, as ε→ 0, of the limit spectral density, as n→∞, of Dε n. The proof is elementary and based on a formula given in [12] for the Cauchy transform of the limit spectral distribution of Dε n as n→∞. For each n, we consider an Hermitian or real symmetric random matrix Xn = [xi,j ] n i,j=1 and a real diagonal matrix Dn = diag(an(1), . . . , an(n)). We suppose that: (a) the entries xi,j of Xn are independent (up to symmetry), centered, Gaussian with variance denoted by σ2 n(i, j), (b) for a certain bounded function σ defined on [0, 1] × [0, 1] and a certain bounded real function f defined on [0, 1], we have, in the L∞ topology, σ n(bnxc, bnyc) −→ n→∞ σ (x, y) and an(bnxc) −→ n→∞ f(x), (c) the set of discontinuities of the function σ is closed and intersects a finite number of times any vertical line of the square [0, 1]2. For ε ≥ 0, let us define, for all n, D n = Dn + √ ε n Xn. Date: April 27, 2011. 2000 Mathematics Subject Classification. 15A52, 46L54.