314 research outputs found
What is a multiple orthogonal polynomial?
This is an extended version of our note in the Notices of the American
Mathematical Society 63 (2016), no. 9, in which we explain what multiple
orthogonal polynomials are and where they appear in various applications.Comment: 5 pages, 2 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
Non-crossing Brownian paths and Dyson Brownian motion under a moving boundary
We compute analytically the probability that a set of Brownian
paths do not cross each other and stay below a moving boundary up to time . We show that for large it decays as a power
law . The decay exponent is obtained
as the ground state energy of a quantum system of non-interacting fermions
in a harmonic well in the presence of an infinite hard wall at position .
Explicit expressions for are obtained in various limits of and
, in particular for large and large . We obtain the joint
distribution of the positions of the walkers in the presence of the moving
barrier at large time. We extend our results to the
case of Dyson Brownian motions (corresponding to the Gaussian Unitary
Ensemble) in the presence of the same moving boundary .
For we show that the system provides a realization of a Laguerre
biorthogonal ensemble in random matrix theory. We obtain explicitly the average
density near the barrier, as well as in the bulk far away from the barrier.
Finally we apply our results to non-crossing Brownian bridges on the
interval under a time-dependent barrier .Comment: 44 pages, 13 figure
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