Related papers: Complete diagrammatics of the single ring theorem
Normalized eigenvalue counting measure of the sum of two Hermitian (or real symmetric) matrices $A_{n}$ and $B_{n}$ rotated independently with respect to each other by the random unitary (or orthogonal) Haar distributed matrix $U_{n}$ (i.e.…
We carry out some algebraic and analytic properties of a new class of orthogonal polyanalytic polynomials, including their operational formulas, recurrence relations, generating functions, integral representations and different…
Every countable directed graph generates a Fock space Hilbert space and a family of partial isometries. These operators also arise from the left regular representations of free semigroupoids derived from directed graphs. We develop a…
We study relations between the eigenvectors of rational matrix functions on the Riemann sphere. Our main result is that for a subclass of functions that are products of two elementary blocks it is possible to represent these relations in a…
For linear operators $L, T$ and nonlinear maps $P$, we describe classes of simple maps $F = I - P T$, $F = L - P$ between Banach and Hilbert spaces, for which no point has more than two preimages. The classes encompass known examples…
Integrable theory is formulated for correlation functions of characteristic polynomials associated with invariant non-Gaussian ensembles of Hermitean random matrices. By embedding the correlation functions of interest into a more general…
We prove that unitary two-dimensional topological field theories are uniquely characterized by $n$ positive real numbers $\lambda _1,\ldots \lambda _n$ which can be regarded as the eigenvalues of a hermitean handle creation operator. The…
The aim of this paper is to develop an approach to obtain self-adjoint extensions of symmetric operators acting on anti-dual pairs. The main advantage of such a result is that it can be applied for structures not carrying a Hilbert space…
A set of Hamiltonians that are not self-adjoint but have the spectrum of the harmonic oscillator is studied. The eigenvectors of these operators and those of their Hermitian conjugates form a bi-orthogonal system that provides a…
In a recent article a generalization of the binomial distribution associated with a sequence of positive numbers was examined. The analysis of the nonnegativeness of the formal expressions was a key-point to allow to give them a statistical…
Complex extension of quantum mechanics and the discovery of pseudo-unitarily invariant random matrix theory has set the stage for a number of applications of these concepts in physics. We briefly review the basic ideas and present…
The $s$-point correlation function of a Gaussian Hermitian random matrix theory, with an external source tuned to generate a multi-critical singularity, provides the intersection numbers of the moduli space for the $p$-th spin curves…
In this paper we initiate the study of real operator monotonicity for functions of tuples of operators, which are multivariate structured maps with a functional calculus called free functions that preserve the order between real parts (or…
For a commutative ring $R$, we exploit localization techniques and point-free topology to give an explicit realization of both the Zariski frame of $R$ (the frame of radical ideals in $R$) and its Hochster dual frame, as lattices in the…
We explore the Hamiltonian operator H=-d^2/dx^2 + z \delta(x) where x is real, \delta(x) is the Dirac delta function, and z is an arbitrary complex coupling constant. For a purely imaginary z, H has a (real) spectral singularity at…
It has been observed that the statistical distribution of the eigenvalues of random matrices possesses universal properties, independent of the probability law of the stochastic matrix. In this article we find the correlation functions of…
Higher order free moments and cumulants, introduced by Collins, Mingo, \'Sniady and Speicher in 2006, describe the fluctuations of unitarily invariant random matrices in the limit of infinite size. The functional relations between their…
The ensemble inter-relations to be considered are special features of classical cases, where the joint eigenvalue probability density can be computed explicitly. Attention will be focussed too on the consequences of these inter-relations,…
Random matrix theory allows one to deduce the eigenvalue spectrum of a large matrix given only statistical information about its elements. Such results provide insight into what factors contribute to the stability of complex dynamical…
Sequences are often conveniently encoded in the form of a generating function depending on a formal variable. This note presents two observations that allow one to draw conclusions about the generated sequence from the generating function.…