109 research outputs found
Products of Complex Rectangular and Hermitian Random Matrices
Products and sums of random matrices have seen a rapid development in the
past decade due to various analytical techniques available. Two of these are
the harmonic analysis approach and the concept of polynomial ensembles. Very
recently, it has been shown for products of real matrices with anti-symmetric
matrices of even dimension that the traditional harmonic analysis on matrix
groups developed by Harish-Chandra et al. needs to be modified when considering
the group action on general symmetric spaces of matrices. In the present work,
we consider the product of complex random matrices with Hermitian matrices, in
particular the former can be also rectangular while the latter has not to be
positive definite and is considered as a fixed matrix as well as a random
matrix. This generalises an approach for products involving the Gaussian
unitary ensemble (GUE) and circumvents the use there of non-compact group
integrals. We derive the joint probability density function of the real
eigenvalues and, additionally, prove transformation formulas for the
bi-orthogonal functions and kernels.Comment: 25 pages, v2: corrections of minor typos and an additional discussion
of Example IV.
Norms on complex matrices induced by random vectors II: extension of weakly unitarily invariant norms
We improve and expand in two directions the theory of norms on complex
matrices induced by random vectors. We first provide a simple proof of the
classification of weakly unitarily invariant norms on the Hermitian matrices.
We use this to extend the main theorem in [7] from exponent to . Our proofs are much simpler than the originals: they do not require
Lewis' framework for group invariance in convex matrix analysis. This
clarification puts the entire theory on simpler foundations while extending its
range of applicability.Comment: 10 page
Szegő's Theorem and Its Descendants: Spectral Theory for L^2 Perturbations of Orthogonal Polynomials
This book presents a comprehensive overview of the sum rule approach to spectral analysis of orthogonal polynomials, which derives from Gábor Szego's classic 1915 theorem and its 1920 extension. Barry Simon emphasizes necessary and sufficient conditions, and provides mathematical background that until now has been available only in journals. Topics include background from the theory of meromorphic functions on hyperelliptic surfaces and the study of covering maps of the Riemann sphere with a finite number of slits removed. This allows for the first book-length treatment of orthogonal polynomials for measures supported on a finite number of intervals on the real line.
In addition to the Szego and Killip-Simon theorems for orthogonal polynomials on the unit circle (OPUC) and orthogonal polynomials on the real line (OPRL), Simon covers Toda lattices, the moment problem, and Jacobi operators on the Bethe lattice. Recent work on applications of universality of the CD kernel to obtain detailed asymptotics on the fine structure of the zeros is also included. The book places special emphasis on OPRL, which makes it the essential companion volume to the author's earlier books on OPUC
Random walks in Dirichlet environment: an overview
Random Walks in Dirichlet Environment (RWDE) correspond to Random Walks in
Random Environment (RWRE) on where the transition probabilities are
i.i.d. at each site with a Dirichlet distribution. Hence, the model is
parametrized by a family of positive weights ,
one for each direction of . In this case, the annealed law is that
of a reinforced random walk, with linear reinforcement on directed edges. RWDE
have a remarkable property of statistical invariance by time reversal from
which can be inferred several properties that are still inaccessible for
general environments, such as the equivalence of static and dynamic points of
view and a description of the directionally transient and ballistic regimes. In
this paper we give a state of the art on this model and several sketches of
proofs presenting the core of the arguments. We also present new computation of
the large deviation rate function for one dimensional RWDE.Comment: 35 page
Counting Finite Magmas
Given a non-negative integer , we establish a formula for the number of
finite magmas on a set with cardinality up to isomorphism. We then
generalize the method to operations with arbitrary finite arity, which yields a
corrected version of Harrison's formula. Moreover, we present the cycle index
as a helpful tool for practical computations and, based on that, we give a
suitable code in Sage with a few generated examples.Comment: references added; abstract adjusted; minor change
Szegő's Theorem and Its Descendants: Spectral Theory for L^2 Perturbations of Orthogonal Polynomials
This book presents a comprehensive overview of the sum rule approach to spectral analysis of orthogonal polynomials, which derives from Gábor Szego's classic 1915 theorem and its 1920 extension. Barry Simon emphasizes necessary and sufficient conditions, and provides mathematical background that until now has been available only in journals. Topics include background from the theory of meromorphic functions on hyperelliptic surfaces and the study of covering maps of the Riemann sphere with a finite number of slits removed. This allows for the first book-length treatment of orthogonal polynomials for measures supported on a finite number of intervals on the real line.
In addition to the Szego and Killip-Simon theorems for orthogonal polynomials on the unit circle (OPUC) and orthogonal polynomials on the real line (OPRL), Simon covers Toda lattices, the moment problem, and Jacobi operators on the Bethe lattice. Recent work on applications of universality of the CD kernel to obtain detailed asymptotics on the fine structure of the zeros is also included. The book places special emphasis on OPRL, which makes it the essential companion volume to the author's earlier books on OPUC
- …