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

Forrester, Peter; Li, Shi-Hao

doi:10.1090/tran/7957

Classical discrete symplectic ensembles on the linear and exponential lattice: skew orthogonal polynomials and correlation functions
HTML articles powered by AMS MathViewer

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

M. Adler, P. J. Forrester, T. Nagao, and P. van Moerbeke, Classical skew orthogonal polynomials and random matrices, J. Statist. Phys. 99 (2000), no. 1-2, 141–170. MR 1762659, DOI 10.1023/A:1018644606835
M. Adler, E. Horozov, and P. van Moerbeke, The Pfaff lattice and skew-orthogonal polynomials, Internat. Math. Res. Notices 11 (1999), 569–588. MR 1695231, DOI 10.1155/S107379289900029X
Kazuhiko Aomoto, On elliptic product formulas for Jackson integrals associated with reduced root systems, J. Algebraic Combin. 8 (1998), no. 2, 115–126. MR 1648460, DOI 10.1023/A:1008629309210
Richard Askey, Some basic hypergeometric extensions of integrals of Selberg and Andrews, SIAM J. Math. Anal. 11 (1980), no. 6, 938–951. MR 595822, DOI 10.1137/0511084
R. Askey and S. K. Suslov, The $q$-harmonic oscillator and the Al-Salam and Carlitz polynomials, Lett. Math. Phys. 29 (1993), no. 2, 123–132. MR 1245186, DOI 10.1007/BF00749728
T. H. Baker and P. J. Forrester, Multivariable Al-Salam & Carlitz polynomials associated with the type $A$ $q$-Dunkl kernel, Math. Nachr. 212 (2000), 5–35. MR 1750712, DOI 10.1002/(SICI)1522-2616(200004)212:1<5::AID-MANA5>3.3.CO;2-J
Alexei Borodin, Determinantal point processes, The Oxford handbook of random matrix theory, Oxford Univ. Press, Oxford, 2011, pp. 231–249. MR 2932631
Alexei Borodin and Leonid Petrov, Integrable probability: from representation theory to Macdonald processes, Probab. Surv. 11 (2014), 1–58. MR 3178541, DOI 10.1214/13-PS225
Alexei Borodin and Eugene Strahov, Correlation kernels for discrete symplectic and orthogonal ensembles, Comm. Math. Phys. 286 (2009), no. 3, 933–977. MR 2472022, DOI 10.1007/s00220-008-0629-8
Richard P. Brent, Christian Krattenthaler, and Ole Warnaar, Discrete analogues of Macdonald-Mehta integrals, J. Combin. Theory Ser. A 144 (2016), 80–138. MR 3534065, DOI 10.1016/j.jcta.2016.06.005
N. G. de Bruijn, On some multiple integrals involving determinants, J. Indian Math. Soc. (N.S.) 19 (1955), 133–151 (1956). MR 79647
James R. Bunch, A note on the stable decompostion of skew-symmetric matrices, Math. Comp. 38 (1982), no. 158, 475–479. MR 645664, DOI 10.1090/S0025-5718-1982-0645664-X
Xiang-Ke Chang, Yi He, Xing-Biao Hu, and Shi-Hao Li, Partial-skew-orthogonal polynomials and related integrable lattices with Pfaffian tau-functions, Comm. Math. Phys. 364 (2018), no. 3, 1069–1119. MR 3875823, DOI 10.1007/s00220-018-3273-y
B. Eynard, Asymptotics of skew orthogonal polynomials, J. Phys. A 34 (2001), no. 37, 7591–7605. MR 1863247, DOI 10.1088/0305-4470/34/37/312
P. J. Forrester, Log-gases and random matrices, London Mathematical Society Monographs Series, vol. 34, Princeton University Press, Princeton, NJ, 2010. MR 2641363, DOI 10.1515/9781400835416
Peter J. Forrester, Meet Andréief, Bordeaux 1886, and Andreev, Kharkov 1882–1883, Random Matrices Theory Appl. 8 (2019), no. 2, 1930001, 9. MR 3943898, DOI 10.1142/S2010326319300018
Peter J. Forrester, Satya N. Majumdar, and Grégory Schehr, Non-intersecting Brownian walkers and Yang-Mills theory on the sphere, Nuclear Phys. B 844 (2011), no. 3, 500–526. MR 2747559, DOI 10.1016/j.nuclphysb.2010.11.013
Peter J. Forrester, Taro Nagao, and Eric M. Rains, Correlation functions for random involutions, Int. Math. Res. Not. , posted on (2006), Art. ID 89796, 35. MR 2233721, DOI 10.1155/IMRN/2006/89796
Peter J. Forrester and Eric M. Rains, Correlations for superpositions and decimations of Laguerre and Jacobi orthogonal matrix ensembles with a parameter, Probab. Theory Related Fields 130 (2004), no. 4, 518–576. MR 2102890, DOI 10.1007/s00440-004-0374-7
Peter J. Forrester and Eric M. Rains, Interpretations of some parameter dependent generalizations of classical matrix ensembles, Probab. Theory Related Fields 131 (2005), no. 1, 1–61. MR 2105043, DOI 10.1007/s00440-004-0375-6
Peter J. Forrester and S. Ole Warnaar, The importance of the Selberg integral, Bull. Amer. Math. Soc. (N.S.) 45 (2008), no. 4, 489–534. MR 2434345, DOI 10.1090/S0273-0979-08-01221-4
Saugata Ghosh, Skew-orthogonal polynomials and random matrix theory, CRM Monograph Series, vol. 28, American Mathematical Society, Providence, RI, 2009. MR 2560290, DOI 10.1090/crmm/028
Mourad E. H. Ismail, Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2005. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey. MR 2191786, DOI 10.1017/CBO9781107325982
Masahiko Ito and Peter J. Forrester, A bilateral extension of the $q$-Selberg integral, Trans. Amer. Math. Soc. 369 (2017), no. 4, 2843–2878. MR 3592530, DOI 10.1090/tran/6851
Kurt Johansson, Discrete orthogonal polynomial ensembles and the Plancherel measure, Ann. of Math. (2) 153 (2001), no. 1, 259–296. MR 1826414, DOI 10.2307/2661375
Kurt Johansson, Non-intersecting paths, random tilings and random matrices, Probab. Theory Related Fields 123 (2002), no. 2, 225–280. MR 1900323, DOI 10.1007/s004400100187
Kevin W. J. Kadell, A proof of Askey’s conjectured $q$-analogue of Selberg’s integral and a conjecture of Morris, SIAM J. Math. Anal. 19 (1988), no. 4, 969–986. MR 946655, DOI 10.1137/0519067
R. Koekoek and R. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its $q$-analogue, arXiv:math/9602214 (1996).
Karl Liechty and Dong Wang, Nonintersecting Brownian motions on the unit circle, Ann. Probab. 44 (2016), no. 2, 1134–1211. MR 3474469, DOI 10.1214/14-AOP998
Tetsuya Masuda, Katsuhisa Mimachi, Yoshiomi Nakagami, Masatoshi Noumi, and Kimio Ueno, Representations of the quantum group $\textrm {SU}_q(2)$ and the little $q$-Jacobi polynomials, J. Funct. Anal. 99 (1991), no. 2, 357–386. MR 1121618, DOI 10.1016/0022-1236(91)90045-7
Taro Nagao and Peter J. Forrester, Vicious random walkers and a discretization of Gaussian random matrix ensembles, Nuclear Phys. B 620 (2002), no. 3, 551–565. MR 1877962, DOI 10.1016/S0550-3213(01)00561-2
A. F. Nikiforov and S. K. Suslov, Classical orthogonal polynomials of a discrete variable on nonuniform lattices, Lett. Math. Phys. 11 (1986), no. 1, 27–34. MR 824673, DOI 10.1007/BF00417461
E. Rains, Correlation functions for symmetrized increasing subsequences, arXiv:0006097 (2000).
Harold Widom, On the relation between orthogonal, symplectic and unitary matrix ensembles, J. Statist. Phys. 94 (1999), no. 3-4, 347–363. MR 1675356, DOI 10.1023/A:1004516918143