On the empirical distribution of eigenvalues of a class of large dimensional random matrices.

*(English)*Zbl 0833.60038The eigenvalue distribution of random matrices
\[
H_N (x,y) = A_N (x,y) + {1\over N} \sum^n_{j = 1} \tau_j \xi_j (x) \xi_j (y), \qquad x,y = \overline{1,N},\tag{1}
\]
where Hermitian \(A_N\), real \(\tau\), and complex \(\xi\) are random is studied in the limit \(n,N \to \infty\), \(n/N \to c> 0\). It is proved that the eigenvalue distribution function (EDF) \(\sigma(\lambda; H_N) \equiv \# \{ \lambda^{(N)}_i \leq \lambda\}N^{-1}\), \(\lambda_i^{(N)}\) are eigenvalues of \(H_N\), converges with probability 1 to a nonrandom \(\sigma (\lambda)\),
\[
\lim_{n,N \to \infty, n/N \to c} \sigma(\lambda; H_N) = \sigma(\lambda), \tag{2}
\]
in the case if \(A_N\), \(\tau\), and \(\xi\) are jointly independent, if there exist nonrandom limits \(\sigma_A (\lambda)\) and \(\sigma_\tau (\lambda)\) of \(\sigma (\lambda; A_N)\) and \(\sigma(\lambda; \text{diag} \{\tau_1, \dots, \tau_n\})\), respectively, and if \(\xi_j(x)\) are independent identically distributed random variables. It is shown that the Stieltjes transform \(m(z) =\)

\(\int(\lambda - z)^{-1} d\sigma (\lambda)\) is a unique solution of the equation \[ m(z) = \int d\sigma_A (\lambda) \left [\lambda - z + c \int {\mu d\sigma_\tau (\mu)\over 1 + \mu m (z)} \right ]^{-1}.\tag{3} \] This result could be regarded as a generalization of the pioneering work of V. Marchenko and L. Pastur [Math. USSR, Sb. 1, 457-483 (1967); translation from Math. Sb., n. Ser. 72(114), 507-536 (1967; Zbl 0152.161)], where the convergence \(\sigma(\lambda; H_N)\to\sigma(\lambda)\) in probability was proved and equation (3) was derived for the ensemble of the form (1) with symmetric nonrandom \(A_N\). However, this generalization is not full in the sense that Marchenko and Pastur have considered the case of arbitrary (not necessarily identically) distributed \(\xi_j(x)\) admitting a dependence that vanishes in the limit \(N \to \infty\). In particular, the vectors \(\vec \xi_j\) can be taken as uniformly distributed over the unit sphere \(\mathbb{C}^N\). In Marchenko-Pastur’s paper the resolvent approach was at first developed in application to the spectral theory of random matrices. This approach is essentially modified in the paper under consideration.

It should be noted that the resolvent technique seems to be one of the most natural ones to investigate limiting EDF’s of large random matrices. It can be used in studies of various ensembles of random matrices with independent entries [V. L. Girko, “Spectral theory of random matrices” (1988; Zbl 0656.15012)] and wide classes of discrete random operators [the reviewer and L. A. Pastur, Commun. Math. Phys. 153, No. 3, 605-646 (1993; Zbl 0772.60046)]. Recent modifications of the method give fairly short proofs of results similar to (2)–(3) even for the cases when random matrix entries have correlations that do not vanish in the limit \(N\to \infty\).

\(\int(\lambda - z)^{-1} d\sigma (\lambda)\) is a unique solution of the equation \[ m(z) = \int d\sigma_A (\lambda) \left [\lambda - z + c \int {\mu d\sigma_\tau (\mu)\over 1 + \mu m (z)} \right ]^{-1}.\tag{3} \] This result could be regarded as a generalization of the pioneering work of V. Marchenko and L. Pastur [Math. USSR, Sb. 1, 457-483 (1967); translation from Math. Sb., n. Ser. 72(114), 507-536 (1967; Zbl 0152.161)], where the convergence \(\sigma(\lambda; H_N)\to\sigma(\lambda)\) in probability was proved and equation (3) was derived for the ensemble of the form (1) with symmetric nonrandom \(A_N\). However, this generalization is not full in the sense that Marchenko and Pastur have considered the case of arbitrary (not necessarily identically) distributed \(\xi_j(x)\) admitting a dependence that vanishes in the limit \(N \to \infty\). In particular, the vectors \(\vec \xi_j\) can be taken as uniformly distributed over the unit sphere \(\mathbb{C}^N\). In Marchenko-Pastur’s paper the resolvent approach was at first developed in application to the spectral theory of random matrices. This approach is essentially modified in the paper under consideration.

It should be noted that the resolvent technique seems to be one of the most natural ones to investigate limiting EDF’s of large random matrices. It can be used in studies of various ensembles of random matrices with independent entries [V. L. Girko, “Spectral theory of random matrices” (1988; Zbl 0656.15012)] and wide classes of discrete random operators [the reviewer and L. A. Pastur, Commun. Math. Phys. 153, No. 3, 605-646 (1993; Zbl 0772.60046)]. Recent modifications of the method give fairly short proofs of results similar to (2)–(3) even for the cases when random matrix entries have correlations that do not vanish in the limit \(N\to \infty\).

Reviewer: A.Khorunzhy (Khar’kov)