Related papers: Derivative principles for invariant ensembles
Loss-induced transmission in waveguides, and reversed pump dependence in lasers, are two prominent examples of counter-intuitive effects in non-Hermitian systems with patterned gain and loss. By analyzing the eigenvalue dynamics of complex…
Conjugation covariants of matrices are applied to study the real algebraic variety consisting of complex Hermitian matrices with a bounded number of distinct eigenvalues. A minimal generating system of the vanishing ideal of degenerate…
To develop a unitary quantum theory with probabilistic description for pseudo- Hermitian systems one needs to consider the theories in a different Hilbert space endowed with a positive definite metric operator. There are different…
In quantum chromodynamics (QCD) at nonzero chemical potential, the eigenvalues of the Dirac operator are scattered in the complex plane. Can the fluctuation properties of the Dirac spectrum be described by universal predictions of…
In probabilistic coherence spaces, a denotational model of probabilistic functional languages, morphisms are analytic and therefore smooth. We explore two related applications of the corresponding derivatives. First we show how derivatives…
In this work we continue the study of the Weyl asymptotics of the distribution of eigenvalues of non-self-adjoint (pseudo)differential operators with small random perturbations, by treating the case of multiplicative perturbations in…
Polynomial ensembles are a sub-class of probability measures within determinantal point processes. Examples include products of independent random matrices, with applications to Lyapunov exponents, and random matrices with an external…
Duality identities in random matrix theory for products and powers of characteristic polynomials, and for moments, are reviewed. The structure of a typical duality identity for the average of a positive integer power $k$ of the…
This paper presents the forward and backward derivatives of partial eigendecomposition, i.e. where it only obtains some of the eigenpairs, of a real symmetric matrix for degenerate cases. The numerical calculation of forward and backward…
The eigenvector-eigenvalue identity relates the eigenvectors of a Hermitian matrix to its eigenvalues and the eigenvalues of its principal submatrices in which the jth row and column have been removed. We show that one-dimensional arrays of…
We consider large Hermitian matrices whose entries are defined by evaluating the exponential function along orbits of the skew-shift $\binom{j}{2} \omega+jy+x \mod 1$ for irrational $\omega$. We prove that the eigenvalue distribution of…
Complex Hermitian random matrices with a unitary symmetry can be distinguished by a weight function. When this is even, it is a known result that the distribution of the singular values can be decomposed as the superposition of two…
We give abstract versions of the large deviation theorem for the distribution of zeros of polynomials and apply them to the characteristic polynomials of Hermitian random matrices. We obtain new estimates related to the local semi-circular…
This note presents explicit formulae for the exponentials of a wide variety of matrices which are 4x4, anti-Hermitian. Easily verifiable conditions characterizing when such matrices admit one of three minimal polynomials are also given.…
Applications of variational methods are typically restricted to conservative systems. Some extensions to dissipative systems have been reported too but require ad hoc techniques such as the artificial doubling of the dynamical variables.…
In this paper we prove large and moderate deviations principles for the recursive kernel estimator of a probability density function and its partial derivatives. Unlike the density estimator, the derivatives estimators exhibit a quadratic…
We investigate traces of powers of random matrices whose distributions are invariant under rotations (with respect to the Hilbert--Schmidt inner product) within a real-linear subspace of the space of $n\times n$ matrices. The matrices we…
Structured perturbation results for invariant subspaces of $\Delta$-Hermitian and Hamiltonian matrices are provided. The invariant subspaces under consideration are associated with the eigenvalues perturbed from a single defective…
We discuss different cases of dissipative Hamiltonian differential-algebraic equations and the linear algebraic systems that arise in their linearization or discretization. For each case we give examples from practical applications. An…
Random matrix models based on an integral over supermatrices are proposed as a natural extension of bosonic matrix models. The subtle nature of superspace integration allows these models to have very different properties from the analogous…