108 research outputs found
The tacnode Riemann-Hilbert problem
The tacnode Riemann-Hilbert problem is a 4 x 4 matrix valued RH problem that
appears in the description of the local behavior of two touching groups of
non-intersecting Brownian motions. The same RH problem was also found by Duits
and Geudens to describe a new critical regime in the two-matrix model.
Delvaux gave integral representations for some of the entries of the 4 x 4
matrix. We complement this work by presenting integral representations for all
of the entries. As a consequence we give an explicit formula for the
Duits-Geudens critical kernel.Comment: 29 pages, 5 figure
Multiple orthogonal polynomial ensembles
Multiple orthogonal polynomials are traditionally studied because of their
connections to number theory and approximation theory. In recent years they
were found to be connected to certain models in random matrix theory. In this
paper we introduce the notion of a multiple orthogonal polynomial ensemble (MOP
ensemble) and derive some of their basic properties. It is shown that Angelesco
and Nikishin systems give rise to MOP ensembles and that the equilibrium
problems that are associated with these systems have a natural interpretation
in the context of MOP ensembles.Comment: 20 pages, no figure
Exceptional Laguerre polynomials
The aim of this paper is to present the construction of exceptional Laguerre
polynomials in a systematic way, and to provide new asymptotic results on the
location of the zeros. To describe the exceptional Laguerre polynomials we
associate them with two partitions. We find that the use of partitions is an
elegant way to express these polynomials and we restate some of their known
properties in terms of partitions. We discuss the asymptotic behavior of the
regular zeros and the exceptional zeros of exceptional Laguerre polynomials as
the degree tends to infinity.Comment: To appear in Studies in Applied Mathematic
- …