ON ASYMPTOTIC EXPANSIONS AND SCALES OF SPECTRAL UNIVERSALITY IN BAND RANDOM MATRIX ENSEMBLES (2008)
BibTeX
@MISC{08onasymptotic,
author = {},
title = {ON ASYMPTOTIC EXPANSIONS AND SCALES OF SPECTRAL UNIVERSALITY IN BAND RANDOM MATRIX ENSEMBLES},
year = {2008}
}
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Abstract
We consider real random symmetric N × N matrices H of the bandtype form with characteristic length b. The matrix entries H(x,y), x ≤ y are independent Gaussian random variables and have the variance proportional to u ( x−y b), where u(t) vanishes at infinity. We study the resolvent G(z) = (H − z) −1, Im z ̸ = 0 in the limit 1 ≪ b ≪ N and obtain explicit expression S(z1, z2) for the leading term of the first correlation function of the normalized trace 〈G(z) 〉 = N −1 Tr G(z). We examine S(λ1+i0, λ2 −i0) on the local scale λ1 −λ2 = r N and show that its asymptotic behavior is determined by the rate of decay of u(t). In particular, if u(t) decays exponentially, then S(r) ∼ −C b 2 N −1 r −3/2. This expression is universal in the sense that the particular form of u determines
