Local spectral universality for random matrices with independent entries

摘要

We consider the local eigenvalue statistics of large self-adjoint $N imes N$ - random matrices, $mathbf{H}=mathbf{H}^*$, with centred independent entries. In contrast to previous works the matrix of variances, $s_{i j}= mathbb{E},|h_{i j}|^2$, is not assumed to be stochastic. Hence the density of states is not the Wigner semicircle law. In this work we prove that as $N$ tends to infinity the $k$ - point correlation function of finitely many eigenvalues becomes universal, i.e., it depends only on the symmetry class of the underlying random matrix ensemble and not on the distributions of its entries. udThe proof consists of three major steps. In the first step we analyse the solution, $mathbf{m}(z)=(m_1(z), dots, m_N(z))$, of the quadratic vector equation (QVE), $-1/m_i(z)= z+ sum_j s_{i j}m_j(z)$, for any complex number $z$. We show that the entries, $m_i$, can be represented as Stieltjes transforms of probability densities on the real line. We characterise these densities in terms of their singularities, which are algebraic of degree at most three. We present a complete stability analysis of the QVE everywhere, including the vicinity of the singularities. This stability analysis is used in the second step. Here we prove that the diagonal elements of the resolvent, $mathbf{G} = (mathbf{H}-z)^{-1}$, satisfy the perturbed QVE, $-1/G_{ii}(z)= z+ sum_j s_{i j}G_{jj}(z)+d_i(z)$, with a random noise vector $mathbf{d}$. We show that as $N$ grows the noise vanishes and the resolvent is close to the deterministic diagonal matrix $ext{diag}(m_1, dots, m_N)$. This result is shown with a precision down to the finest spectral scale, just above the typical eigenvalue spacing. It thus implies the local law and rigidity of the eigenvalue positions for this random matrix model. In the third and final step, we use the Dyson-Brownian-motion to establish universality of the local eigenvalue statistics.