1. CJM 2012 (vol 65 pp. 600)
|Christoffel Functions and Universality in the Bulk for Multivariate Orthogonal Polynomials|
We establish asymptotics for Christoffel functions associated with multivariate orthogonal polynomials. The underlying measures are assumed to be regular on a suitable domain - in particular this is true if they are positive a.e. on a compact set that admits analytic parametrization. As a consequence, we obtain asymptotics for Christoffel functions for measures on the ball and simplex, under far more general conditions than previously known. As another consequence, we establish universality type limits in the bulk in a variety of settings.
Keywords:orthogonal polynomials, random matrices, unitary ensembles, correlation functions, Christoffel functions
Categories:42C05, 42C99, 42B05, 60B20
2. CJM 2011 (vol 64 pp. 805)
|Quantum Random Walks and Minors of Hermitian Brownian Motion|
Considering quantum random walks, we construct discrete-time approximations of the eigenvalues processes of minors of Hermitian Brownian motion. It has been recently proved by Adler, Nordenstam, and van Moerbeke that the process of eigenvalues of two consecutive minors of a Hermitian Brownian motion is a Markov process; whereas, if one considers more than two consecutive minors, the Markov property fails. We show that there are analog results in the noncommutative counterpart and establish the Markov property of eigenvalues of some particular submatrices of Hermitian Brownian motion.
Keywords:quantum random walk, quantum Markov chain, generalized casimir operators, Hermitian Brownian motion, diffusions, random matrices, minor process
Categories:46L53, 60B20, 14L24