## ON ASYMPTOTIC EXPANSIONS AND SCALES OF SPECTRAL UNIVERSALITY IN BAND RANDOM MATRIX ENSEMBLES (2000)

Citations: | 5 - 0 self |

### BibTeX

@MISC{Khorunzhy00onasymptotic,

author = {A. Khorunzhy},

title = {ON ASYMPTOTIC EXPANSIONS AND SCALES OF SPECTRAL UNIVERSALITY IN BAND RANDOM MATRIX ENSEMBLES},

year = {2000}

}

### OpenURL

### Abstract

We consider the family of ensembles of random symmetric N × N matrices H (N,b) of the band-type structure with characteristic length b. The variance of the matrix entries H (N,b) (x, y) is proportional to u ( x−y b) with certain decaying function u(t) ≥ 0. In the limit of (relatively) narrow band width 1 ≪ b ≪ N, we derive explicit expressions for the first terms of 1/b-expansions of the average of the Green function N −1 Tr(H (N,b) −z) −1 and its correlation function as well. The expressions obtained show that there exist several scales of the universal forms of the spectral correlation function. These scales are determined by the rate of decrease of the function u(t). They coincide with those detected in theoretical physics for the localization length and density-density correlator in the band-type random matrix ensembles. 1 Problem, motivation and results

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Citation Context ...n interpreted as the sign of (relatively) localized eigenvectors of H (n,b) in the limit n, b → ∞, with the localization length b 2 /n. Let us note that the case of (6.10) has been also considered in =-=[32]-=- and (6.11) appeared there. In that paper corrections to the average of the Green function were studied also. However it should be stressed that no explicit expressions like (3.2), (3.3) and (3.6), (3... |

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Citation Context ...tified in the sense that one can consider the smoothed eigenvalue density given by Im〈GN(λ + iN −γ )〉 [19] and obtain asymptotics (4.4) its correlation function CN(θ1 + iN −γ , θ2 − iN γ ) with 0 < γ =-=[5]-=-. Regarding this experience, we expect that our results of the next subsection remain valid also in this more realistic regime. 286.2 Scales of asymptotic behaviour Theorem 6.1. Let S(z1, z2) is give... |