237 research outputs found
Recurrence relations and vector equilibrium problems arising from a model of non-intersecting squared Bessel paths
In this paper we consider the model of non-intersecting squared Bessel
processes with parameter , in the confluent case where all particles
start, at time , at the same positive value , remain positive, and
end, at time , at the position . The positions of the paths have a
limiting mean density as which is characterized by a vector
equilibrium problem. We show how to obtain this equilibrium problem from
different considerations involving the recurrence relations for multiple
orthogonal polynomials associated with the modified Bessel functions.
We also extend the situation by rescaling the parameter , letting it
increase proportionally to as increases. In this case we also analyze
the recurrence relation and obtain a vector equilibrium problem for it.Comment: 28 pages, 10 figure
Universality for eigenvalue correlations at the origin of the spectrum
We establish universality of local eigenvalue correlations in unitary random
matrix ensembles (1/Z_n) |\det M|^{2\alpha} e^{-n\tr V(M)} dM near the origin
of the spectrum. If V is even, and if the recurrence coefficients of the
orthogonal polynomials associated with |x|^{2\alpha} e^{-nV(x)} have a regular
limiting behavior, then it is known from work of Akemann et al., and Kanzieper
and Freilikher that the local eigenvalue correlations have universal behavior
described in terms of Bessel functions. We extend this to a much wider class of
confining potentials V. Our approach is based on the steepest descent method of
Deift and Zhou for the asymptotic analysis of Riemann-Hilbert problems. This
method was used by Deift et al. to establish universality in the bulk of the
spectrum. A main part of the present work is devoted to the analysis of a local
Riemann-Hilbert problem near the origin.Comment: 28 pages, 6 figures, technical problem in second version removed, to
appear in Commun. Math. Phy
A Christoffel-Darboux formula for multiple orthogonal polynomials
Bleher and Kuijlaars recently showed that the eigenvalue correlations from
matrix ensembles with external source can be expressed by means of a kernel
built out of special multiple orthogonal polynomials. We derive a
Christoffel-Darboux formula for this kernel for general multiple orthogonal
polynomials. In addition, we show that the formula can be written in terms of
the solution of the Riemann-Hilbert problem for multiple orthogonal
polynomials, which will be useful for asymptotic analysis.Comment: 11 pages, no figure
The local universality of Muttalib-Borodin biorthogonal ensembles with parameter
The Muttalib-Borodin biorthogonal ensemble is a probability density function
for particles on the positive real line that depends on a parameter
and an external field . For we find the large
behavior of the associated correlation kernel with only few restrictions on
. The idea is to relate the ensemble to a type II multiple orthogonal
polynomial ensemble that can in turn be related to a
Riemann-Hilbert problem which we then solve with the Deift-Zhou steepest
descent method. The main ingredient is the construction of the local parametrix
at the origin, with the help of Meijer G-functions, and its matching condition
with a global parametrix. We will present a new iterative technique to obtain
the matching condition, which we expect to be applicable in more general
situations as well.Comment: 51 page
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