Classical discrete symplectic ensembles on the linear and exponential lattice: skew orthogonal polynomials and correlation functions

Abstract

The eigenvalue probability density function for symplectic invariant random matrix ensembles can be generalised to discrete settings involving either a linear or exponential lattice. The corresponding correlation functions can be expressed in terms of certain discrete, and $q$, skew orthogonal polynomials respectively. We give a theory of both of these classes of polynomials, and the correlation kernels determining the correlation functions, in the cases that the weights for the corresponding discrete unitary ensembles are classical. Crucial for this are certain difference operators which relate the relevant symmetric inner products to the skew symmetric ones, and have a tridiagonal action on the corresponding (discrete or $q$) orthogonal polynomials.

arXiv comment31 pages; comments are welcome

References in corpus (7)

• The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue
• Non-intersecting Brownian walkers and Yang-Mills theory on the sphere
• Classical skew orthogonal polynomials and random matrices
• Correlation functions for symmetrized increasing subsequences
• The q-harmonic oscillator and the Al-Salam and Carlitz polynomials
• Correlation Kernels for Discrete Symplectic and Orthogonal Ensembles
• Discrete analogues of Macdonald-Mehta integrals