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

 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. Received: 8 April 2000 / Accepted: 7 June 2002 Published online: 21 October 2002 RID="*" ID="*" Present address: Département de Mathématiques, Université de Versailles Saint-Quentin, 78035 Versailles, France.     

Limits of infinite interaction radius,dimensionality and the number of components for random operators with off-diagonal randomness

We consider random one-body operators that are analogs of the statistical mechanics Hamiltonians with a varying interaction radiusR, the dimensionality of spaced and the number of the field components (orbitals)n. We prove that all the moments of the Green functions for nonreal energies of these operators converge asR, d, n to the products of the average Green functions, just as in the mean field approximation of statistical mechanics. We find in particular the selfconsistent equation for the limiting integrated density of states and the limiting form of the conductivity, which is nonzero on the whole support of the integrated density of states.     

Asymptotic Distribution of Eigenvalues of Weakly Dilute Wishart Matrices

We study the eigenvalue distribution of large random matrices that are randomly diluted. We consider two random matrix ensembles that in the pure (nondilute) case have a limiting eigenvalue distribution with a singular component at the origin. These include the Wishart random matrix ensemble and Gaussian random matrices with correlated entries. Our results show that the singularity in the eigenvalue distribution is rather unstable under dilution and that even weak dilution destroys it. This revised version was published online in July 2006 with corrections to the Cover Date.