On Asymptotic Expansions and Scales of Spectral Universality in Band Random Matrix Ensembles

Abstract:

 We consider real random symmetric N × N matrices H of the band-type form with characteristic length b. The matrix entries \(\) are independent Gaussian random variables and have the variance proportional to \(\), where u(t) vanishes at infinity. We study the resolvent \(\) in the limit \(\) and obtain the explicit expression \(\) for the leading term of the first correlation function of the normalized trace \(\).

We examine \(\) on the local scale \(\) and show that its asymptotic behavior is determined by the rate of decay of u(t). In particular, if u(t) decays exponentially, then \(\). This expression is universal in the sense that the particular form of u determines the value of C > 0 only. Our results agree with those detected in both numerical and theoretical physics studies of spectra of band random matrices.