The Circular Law: ten years later

The V-density of eigenvalues of non symmetric random matrices and rigorous proof of the strong Circular law

“…To avoid mistakes in investigations of the spectra of random matrices based on the V-transform we repeat here the strong proof of corresponding results (see [74][75][76][77][78][79][80][81][82][83][84][85][86][87][88][89]) and explain how regularization of random determinant must be performed on the base of the V-transform. The unsolved problems in the spectral theory of nonselfadjoint random matrices are the general description of their normalized spectral functions under the assumption that the entries of the random matrices are independent, infinitesimal, and the dimension of the matrices tends to infinity.…”

“…In [49] (1991) it was explained that the expected normalized spectral function of H n tends to the circular law, when the entries of the matrix H n are iid real normal random variables. In [74][75][76][77][78][79][80][81][82][83][84][85][86][87][88][89] by means of the V-transform, the study of the spectral functions of nonselfadjoint random matrices was reduced to that of the spectral functions of some Hermitian matrices. It was shown in 1984-1985 [75][76][77][78][79][80] that the so-called Strong Circular and Strong Elliptic Laws for certain random matrices are valid.…”

“…In [84][85][86] the Circular Law without the condition that ξ\^ has density was proved, but instead of the condition sup n max,-,j = i ..... n Ε|ξ^η ) ν /ή| 2+<51 < c < oo we require < c < oo.…”

“…The last step consists in application for this integral the V-transform [82, ρ .433]. Note that if the densities of the enties of random matrices do not exist, then the proof of the Circular Law is much more difficult (see [84]). …”

“…Much difficult to prove this Law for matrices with arbitrary independent entries. It was done in [84]. The possible limit spectral densities were of n.s.f.…”