Limiting eigenvalue distribution for band random matrices

LIMITING EIGENVALUE DISTRIBUTION FOR BAND RANDOM MATRICES S. A. Molchanov, L. A. Pastur, and A. M. Khorunzhii An equation is obtained for the Stieltjes transform of the normalized eigenvalue distribution of band random matrices in the limit in which the band width and rank of the matrix simultaneously tend to infinity. Conditions under which this limit agrees with the semicircle law are found. 1.

STATEMENT

OF PROBLEM,

FORMULATION

OF RESULTS, AND DISCUSSION

Random symmetric, or Hermitian matrices arise in many branches of physics and mathematics (see, for example, the reviews [1--5] and the references there). Among the many and numerous problems associated with the properties of Such matrices, one of the simplest and, at the same time, most important is that of the distribution of their eigenva!ues. As was first shown by Wigner [6], in the case of matrices with independent Gaussian elements this problem can be solved exactly in the limit of infinite rank of these matrices. Namely, let the symmetric matrix W(n) of rank n = 2m + 1 have elements of the form

W(~)(x, y)=n-'Z2V(x, y), where

V(x, y) = V(y, x)

[xl, ly]
a~=-~-~v ~(t)dt,

a2=

u2(t)dt.

Hence and from the Cauchy--Schwarz inequality we obtain the rigorous inequality

a~Zaz-i