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

Abstract

We review some results obtained in series of papers on non Hermitian random matrices in some problems of spin glasses and neural nets. We present new theory of such matrices on the basis of the V-transform of normalized spectral function (n.s.f.) i^n(^,2/) of the eigenvalues of non symmetric matrix Ξ with n.s.f. /x n (x,r) of the eigenvalues of the Hermitian G-matrix (Ξ -τ/) (Ξ -τ/)*, τ = t + is : 7 q φ. Ο, and the modified V^-transform:where ε > 0. This article discusses methodological approach which allows one to obtain rigorous proof of the strong Circular law and to describe the region where the eigenvalues of large non Hermitian random matrices are distributed.

Strong Global Law and the Sombrero probability density

“…THEOREM 1 [6, p.94]. For every n, let the random entries ξ\"\ i = l, ...,n, j = l,..., n, of the complex matrix Ξ ηχη = (ί|" )ί=ί,'...,η be independent, Then, wifch probabilifcy one, for almost all <c,t and s(see [6, p.447]) lim |μ η (χ, ί, s) -.P n (.τ, t, 5) | = 0, n->oo (4) where F n (x,£,s) is the distribution function whose Stielt/es fcransform is given by the formula ' n (r)(Q^(u,t,s)r l S' n (r)} ,« = z + uj, (5) S"(T) = A' 1 [C" -r/n]^-1 , A n = , B n = t=l, ... ,n , C" = are diagonal matrices satisfying the System of V-equations K^s [6, p.94]:…”

“…Thus, in view of the inequality for the rth absolute moments of a sum of random variables (martingale differences) (see [1][2][3][4][5] where ε > 0. By choosing proper small £ 2 , £3, and ε > 0, we complete the proof of Theorem 8.…”

Numerical and Monte Carlo verification of V-distribution

“…In this article we continue to develop new V-analysis (see [60][61][62][63][64][65][66][67][68][69][70][71][72][73][74][75]82) for such matrices and give the domain where its eigenvalues are distributed. Therefore, the main purpose of this article consists to attract physicists to new analysis of random non symmetrical matrices which appear in many modern problems.…”

“…The most important result of articles [71,72], [82] is the V-relation for normalized spectral function (n.s.f.) of the eigenvalues of non symmetric matrix Ξ with n.s.f.…”

“…(see V -equation in [72]). Let Ξ = (£ο')Γ·=ι ^e a random complex matrix, whose entries ξ^ are independent for every n, given on the common probability space, Girko also has considered a particular special case of random matrices, when there exists simple formula for this density, when the matrix A = (o,iSij)^nl is a diagonal matrix with diagonal entries: a; = a + i6, z = l,...,n; a, = -a-i&, i = 1-f η, ...,2η and entries of matrix Ξ are independent, their expectation equal zero and variances equal n-' [82].…”