3,007 research outputs found
Average characteristic polynomials for multiple orthogonal polynomial ensembles
Multiple orthogonal polynomials (MOP) are a non-definite version of matrix
orthogonal polynomials. They are described by a Riemann-Hilbert matrix Y
consisting of four blocks Y_{1,1}, Y_{1,2}, Y_{2,1} and Y_{2,2}. In this paper,
we show that det Y_{1,1} (det Y_{2,2}) equals the average characteristic
polynomial (average inverse characteristic polynomial, respectively) over the
probabilistic ensemble that is associated to the MOP. In this way we generalize
classical results for orthogonal polynomials, and also some recent results for
MOP of type I and type II. We then extend our results to arbitrary products and
ratios of characteristic polynomials. In the latter case an important role is
played by a matrix-valued version of the Christoffel-Darboux kernel. Our proofs
use determinantal identities involving Schur complements, and adaptations of
the classical results by Heine, Christoffel and Uvarov.Comment: 32 page
Average Characteristic Polynomials of Determinantal Point Processes
We investigate the average characteristic polynomial where the 's are real random variables
which form a determinantal point process associated to a bounded projection
operator. For a subclass of point processes, which contains Orthogonal
Polynomial Ensembles and Multiple Orthogonal Polynomial Ensembles, we provide a
sufficient condition for its limiting zero distribution to match with the
limiting distribution of the random variables, almost surely, as goes to
infinity. Moreover, such a condition turns out to be sufficient to strengthen
the mean convergence to the almost sure one for the moments of the empirical
measure associated to the determinantal point process, a fact of independent
interest. As an application, we obtain from a theorem of Kuijlaars and Van
Assche a unified way to describe the almost sure convergence for classical
Orthogonal Polynomial Ensembles. As another application, we obtain from
Voiculescu's theorems the limiting zero distribution for multiple Hermite and
multiple Laguerre polynomials, expressed in terms of free convolutions of
classical distributions with atomic measures.Comment: 26 page
On Certain Wronskians of Multiple Orthogonal Polynomials
We consider determinants of Wronskian type whose entries are multiple orthogonal polynomials associated with a path connecting two multi-indices. By assuming that the weight functions form an algebraic Chebyshev (AT) system, we show that the polynomials represented by the Wronskians keep a constant sign in some cases, while in some other cases oscillatory behavior appears, which generalizes classical results for orthogonal polynomials due to Karlin and Szegő. There are two applications of our results. The first application arises from the observation that the m-th moment of the average characteristic polynomials for multiple orthogonal polynomial ensembles can be expressed as a Wronskian of the type II multiple orthogonal polynomials. Hence, it is straightforward to obtain the distinct behavior of the moments for odd and even m in a special multiple orthogonal ensemble - the AT ensemble. As the second application, we derive some Turán type inequalities for multiple Hermite and multiple Laguerre polynomials (of two kinds). Finally, we study numerically the geometric configuration of zeros for the Wronskians of these multiple orthogonal polynomials. We observe that the zeros have regular configurations in the complex plane, which might be of independent interest
The averaged characteristic polynomial for the Gaussian and chiral Gaussian ensembles with a source
In classical random matrix theory the Gaussian and chiral Gaussian random
matrix models with a source are realized as shifted mean Gaussian, and chiral
Gaussian, random matrices with real , complex ( and
real quaternion ) elements. We use the Dyson Brownian motion model
to give a meaning for general . In the Gaussian case a further
construction valid for is given, as the eigenvalue PDF of a
recursively defined random matrix ensemble. In the case of real or complex
elements, a combinatorial argument is used to compute the averaged
characteristic polynomial. The resulting functional forms are shown to be a
special cases of duality formulas due to Desrosiers. New derivations of the
general case of Desrosiers' dualities are given. A soft edge scaling limit of
the averaged characteristic polynomial is identified, and an explicit
evaluation in terms of so-called incomplete Airy functions is obtained.Comment: 21 page
Polynomial Ensembles and Recurrence Coefficients
Polynomial ensembles are determinantal point processes associated with (non
necessarily orthogonal) projections onto polynomial subspaces. The aim of this
survey article is to put forward the use of recurrence coefficients to obtain
the global asymptotic behavior of such ensembles in a rather simple way. We
provide a unified approach to recover well-known convergence results for real
OP ensembles. We study the mutual convergence of the polynomial ensemble and
the zeros of its average characteristic polynomial; we discuss in particular
the complex setting. We also control the variance of linear statistics of
polynomial ensembles and derive comparison results, as well as asymptotic
formulas for real OP ensembles. Finally, we reinterpret the classical algorithm
to sample determinantal point processes so as to cover the setting of
non-orthogonal projection kernels. A few open problems are also suggested.Comment: 23 page
Painleve IV and degenerate Gaussian Unitary Ensembles
We consider those Gaussian Unitary Ensembles where the eigenvalues have
prescribed multiplicities, and obtain joint probability density for the
eigenvalues. In the simplest case where there is only one multiple eigenvalue
t, this leads to orthogonal polynomials with the Hermite weight perturbed by a
factor that has a multiple zero at t. We show through a pair of ladder
operators, that the diagonal recurrence coefficients satisfy a particular
Painleve IV equation for any real multiplicity. If the multiplicity is even
they are expressed in terms of the generalized Hermite polynomials, with t as
the independent variable.Comment: 17 page
A note on biorthogonal ensembles
We consider ensembles of random matrices, known as biorthogonal ensembles,
whose eigenvalue probability density function can be written as a product of
two determinants. These systems are closely related to multiple orthogonal
functions. It is known that the eigenvalue correlation functions of such
ensembles can be written as a determinant of a kernel function. We show that
the kernel is itself an average of a single ratio of characteristic
polynomials. In the same vein, we prove that the type I multiple polynomials
can be expressed as an average of the inverse of a characteristic polynomial.
We finally introduce a new biorthogonal matrix ensemble, namely the chiral
unitary perturbed by a source term.Comment: 20 page
Correlation kernels for sums and products of random matrices
Let be a random matrix whose squared singular value density is a
polynomial ensemble. We derive double contour integral formulas for the
correlation kernels of the squared singular values of and , where
is a complex Ginibre matrix and is a truncated unitary matrix. We also
consider the product of and several complex Ginibre/truncated unitary
matrices. As an application, we derive the precise condition for the squared
singular values of the product of several truncated unitary matrices to follow
a polynomial ensemble. We also consider the sum where is a GUE
matrix and is a random matrix whose eigenvalue density is a polynomial
ensemble. We show that the eigenvalues of follow a polynomial ensemble
whose correlation kernel can be expressed as a double contour integral. As an
application, we point out a connection to the two-matrix model.Comment: 33 pages, some changes suggested by the referee is made and some
references are adde
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