9,179 research outputs found
Mehler-Heine asymptotics for multiple orthogonal polynomials
Mehler-Heine asymptotics describe the behavior of orthogonal polynomials near
the edges of the interval where the orthogonality measure is supported. For
Jacobi polynomials and Laguerre polynomials this asymptotic behavior near the
hard edge involves Bessel functions . We show that the asymptotic
behavior near the endpoint of the interval of (one of) the measures for
multiple orthogonal polynomials involves a generalization of the Bessel
function. The multiple orthogonal polynomials considered are Jacobi-Angelesco
polynomials, Jacobi-Pi\~neiro polynomials, multiple Laguerre polynomials,
multiple orthogonal polynomials associated with modified Bessel functions (of
the first and second kind), and multiple orthogonal polynomials associated with
Meijer -functions.Comment: 15 pages. Typos corrected, references updated, section "concluding
remarks" adde
Universal random matrix correlations of ratios of characteristic polynomials at the spectral edges
It has been shown recently by Fyodorov and Strahov [math-ph/0204051] that Cauchy transforms of orthogonal polynomials appear naturally in general correlation functions containing ratios of characteristic polynomials of random N×N Hermitian matrices. Our main goal is to investigate the issue of universality of large N asymptotics for those Cauchy transforms for a wide class of weight functions. Our analysis covers three different scaling regimes: the “hard edge”, the “bulk” and the “soft edge” of the spectrum, thus extending the earlier results known for the bulk. The principal tool is to show that for finite matrix size N the auxiliary “wave functions” associated with the Cauchy transforms obey the same second order differential equation as those associated with the orthogonal polynomials themselves
A Central Limit Theorem for Random Walks on the Dual of a Compact Grassmannian
We consider compact Grassmann manifolds over the real, complex or
quaternionic numbers whose spherical functions are Heckman-Opdam polynomials of
type . From an explicit integral representation of these polynomials we
deduce a sharp Mehler-Heine formula, that is an approximation of the
Heckman-Opdam polynomials in terms of Bessel functions, with a precise estimate
on the error term. This result is used to derive a central limit theorem for
random walks on the semi-lattice parametrizing the dual of , which are
constructed by successive decompositions of tensor powers of spherical
representations of . The limit is the distribution of a Laguerre ensemble in
random matrix theory. Most results of this paper are established for a larger
continuous set of multiplicity parameters beyond the group cases
A Limit Relation for Dunkl-Bessel Functions of Type A and B
We prove a limit relation for the Dunkl-Bessel function of type with
multiplicity parameters on the roots and on where tends to infinity and the arguments are suitably scaled. It
gives a good approximation in terms of the Dunkl-type Bessel function of type
with multiplicity . For certain values of an improved
estimate is obtained from a corresponding limit relation for Bessel functions
on matrix cones.Comment: This is a contribution to the Special Issue on Dunkl Operators and
Related Topics, published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
- …