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

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- by Peter J. Forrester and Shi-Hao Li PDF
- Trans. Amer. Math. Soc.
**373**(2020), 665-698 Request permission

## Abstract:

The eigenvalue probability density function for symplectic invariant random matrix ensembles can be generalized to discrete settings involving either a linear or an 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 in which 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.## References

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## Additional Information

**Peter J. Forrester**- Affiliation: School of Mathematical and Statistics, ARC Centre of Excellence for Mathematical and Statistical Frontiers, The University of Melbourne, Victoria 3010, Australia
- MR Author ID: 68170
- Email: pjforr@unimelb.edu.au
**Shi-Hao Li**- Affiliation: School of Mathematical and Statistics, ARC Centre of Excellence for Mathematical and Statistical Frontiers, The University of Melbourne, Victoria 3010, Australia
- MR Author ID: 1224802
- Email: shihao.li@unimelb.edu.au
- Received by editor(s): February 25, 2019
- Received by editor(s) in revised form: July 18, 2019
- Published electronically: September 23, 2019
- Additional Notes: The first author acknowledges partial support from ARC grant DP170102028.

This work was part of a research program supported by the Australian Research Council (ARC) through the ARC Centre of Excellence for Mathematical and Statistical frontiers (ACEMS) - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**373**(2020), 665-698 - MSC (2010): Primary 15B52; Secondary 33E20, 15A15
- DOI: https://doi.org/10.1090/tran/7957
- MathSciNet review: 4042888