On asymptotic expansions and scales of spectral universality in band random matrix ensembles

We consider real random symmetric N x N matrices H of the band-type form with characteristic length b. The matrix entries H(x, y), x less than or equal to 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 6(z) = (H - z)(-1), Im z not equal 0 in the limit 1 much less than b much less than N and obtain the explicit expression S(z(1), z(2)) for the leading term of the first correlation function of the normalized trace <G (z)> = N-1 Tr G(z). We examine S(lambda(1) + i0, lambda(2) - i0) on the local scale lambda(1) - lambda(2) = 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) similar to -C b(2)N(-1)r(-3/2). 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.