Universal relation between Green functions in random matrix theory

Abstract

We prove that in random matrix theory there exists a universal relation between the one-point Green's function G and the connected two-point Green's function G c given byThis relation is universal in the sense that it does not depend on the probability distribution of the random matrices for a broad class of distributions, even though G is known to depend on the probability distribution in detail. The universality discussed here represents a different statement than the universality we discovered a couple of 1 years ago, which states that a 2 G c (az, aw) is independent of the probability distribution, where a denotes the width of the spectrum and depends sensitively on the probability distribution. It is shown that the universality proved here also holds for the more general problem of a Hamiltonian consisting of the sum of a deterministic term and a random term analyzed perturbatively

Correlations of nearby levels induced by a random potential

“…They are based either on orthogonal polynomials [2], or on summing over planar diagrams [3,4], or solving an integral equation [5,6]; however the property a) is known only through the orthogonal polynomials approach [2]. For the generalization that we have in view in this article, in which the "unperturbed" part of the Hamiltonian is deterministic, if again for b) a diagrammatic approach still works [3,4,7,8], we are not aware of any method which would allow us to study whether a) still holds. To this effect we shall generalize a method, introduced by Kazakov [9], to the study of correlation functions.…”

Finite Dimensional Statistical Inference

“…The idea is to draw them using Feynman rules derived from a generating function, and perform a resummation of all relevant graphs where averaging over matrices corresponds to connecting in all possible ways the different lines seperately. In many cases, in the large N -limit, only terms with non-crossing lines survive, A general description is proposed in [22], [23], [24]. The nomenclature we use for stating our results deviate some from that found in the literature.…”

Probing non-orthogonality of eigenvectors in non-Hermitian matrix models: diagrammatic approach

“…It will become clear later that K encodes correlations of eigenvectors and L of eigenvalues. These two possible contractions define two different classes of planar [52,53] and K is given as a sum of ladder diagrams. b) An example of a diagram which contributes to L but is subleading in the calculation of K. c) An example of a diagram appearing during the calculation of L, which despite its planarity is subleading.…”