71 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
The hard edge tacnode process and the hard edge Pearcey process with non-intersecting squared Bessel paths
A system of non-intersecting squared Bessel processes is considered which all
start from one point and they all return to another point. Under the scaling of
the starting and ending points when the macroscopic boundary of the paths
touches the hard edge, a limiting critical process is described in the
neighbourhood of the touching point which we call the hard edge tacnode
process. We derive its correlation kernel in an explicit new form which
involves Airy type functions and operators that act on the direct sum of
and a finite dimensional space. As the starting points of
the squared Bessel paths are set to 0, a cusp in the boundary appears. The
limiting process is described near the cusp and it is called the hard edge
Pearcey process. We compute its multi-time correlation kernel which extends the
existing formulas for the single-time kernel. Our pre-asymptotic correlation
kernel involves the ratio of two Toeplitz determinants which are rewritten
using a Borodin-Okounkov type formula.Comment: 49 pages, 4 figure
An equilibrium problem for the limiting eigenvalue distribution of rational Toeplitz matrices
We consider the asymptotic behavior of the eigenvalues of Toeplitz matrices
with rational symbol as the size of the matrix goes to infinity. Our main
result is that the weak limit of the normalized eigenvalue counting measure is
a particular component of the unique solution to a vector equilibrium problem.
Moreover, we show that the other components describe the limiting behavior of
certain generalized eigenvalues. In this way, we generalize the recent results
of Duits and Kuijlaars for banded Toeplitz matrices.Comment: 20 pages, 2 figure
High order three-term recursions, Riemann-Hilbert minors and Nikishin systems on star-like sets
We study monic polynomials generated by a high order three-term
recursion with arbitrary and
for all . The recursion is encoded by a two-diagonal Hessenberg
operator . One of our main results is that, for periodic coefficients
and under certain conditions, the are multiple orthogonal polynomials
with respect to a Nikishin system of orthogonality measures supported on
star-like sets in the complex plane. This improves a recent result of
Aptekarev-Kalyagin-Saff where a formal connection with Nikishin systems was
obtained in the case when .
An important tool in this paper is the study of "Riemann-Hilbert minors", or
equivalently, the "generalized eigenvalues" of the Hessenberg matrix . We
prove interlacing relations for the generalized eigenvalues by using totally
positive matrices. In the case of asymptotically periodic coefficients ,
we find weak and ratio asymptotics for the Riemann-Hilbert minors and we obtain
a connection with a vector equilibrium problem. We anticipate that in the
future, the study of Riemann-Hilbert minors may prove useful for more general
classes of multiple orthogonal polynomials.Comment: 59 pages, 3 figure
- …