Abstract
1. Gaussian random band matrices. Let g ij , 1 ≤ i, j ≤ n, k = 1, . . . , r be independent centered complex Gaussian random variables of covariance 1 n (1 + δij) (i.e., the expectation E(g ijg k ij) = 1 n (1 + δij)), such that g k ij = g k ji. Let for each i, j, k, σn(i, j; k) be a positive real number, such that σn(i, j; k) = σn(j, i; k). Consider the r-tuple of n × n matrices Gn(k) = (σn(i, j; k)g k ij)ij , k = 1, . . . , r. The family {Gn(k)}k is called a family of independent Gaussian random band matrices. As an example, assume that σn(i, j; k) = 1 if |i − j| < cn, and is zero otherwise. In this case all entries of Gn(k) are zero outside a band around the diagonal; this is the reason such matrices are called band matrices. In the case that σn(i, j; k) is identically 1 one recovers the so-called Gaussian Unitary Ensemble, see e.g. [9].
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