Limiting Eigenvalue Distribution of Random Matrices with Correlated Entries

#### A. Boutet de Monvel, A. Khorunzhy, V. Vasilchuk

1996, v.2, №4, 607-636

ABSTRACT

We study the normalised eigenvalue counting function $\sigma_N(\lambda)$ of a random matrix $H_N(p)$ defined as the sum of $p$ rank-one projections onto $N$-dimensional vectors $\vec\xi_j$. We consider the case that the vector components $N^{-1/2} \xi_j(x)$, $x=\overline{1,N}$, $j=\overline{1,N}$, are Gaussian correlated random variables. We derive an integral equation determining $\sigma_N(\lambda )$ in the limit $N,p \to \infty$, and $p/N\to c> 0$. Using this equation, we show that the correlations between the $\xi_j(x)$ can considerably change the analytic structure of the limiting eigenvalue distribution function.

Keywords: random matrices,correlated entries,eigenvalue distribution