Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 079, 36 pages      arXiv:1703.03914      https://doi.org/10.3842/SIGMA.2017.079
Contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications

Elliptic Determinantal Processes and Elliptic Dyson Models

Makoto Katori
Department of Physics, Faculty of Science and Engineering, Chuo University, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan

Received April 19, 2017, in final form September 29, 2017; Published online October 04, 2017

Abstract
We introduce seven families of stochastic systems of interacting particles in one-dimension corresponding to the seven families of irreducible reduced affine root systems. We prove that they are determinantal in the sense that all spatio-temporal correlation functions are given by determinants controlled by a single function called the spatio-temporal correlation kernel. For the four families ${A}_{N-1}$, ${B}_N$, ${C}_N$ and ${D}_N$, we identify the systems of stochastic differential equations solved by these determinantal processes, which will be regarded as the elliptic extensions of the Dyson model. Here we use the notion of martingales in probability theory and the elliptic determinant evaluations of the Macdonald denominators of irreducible reduced affine root systems given by Rosengren and Schlosser.

Key words: elliptic determinantal processes; elliptic Dyson models; determinantal martingales; elliptic determinant evaluations; irreducible reduced affine root systems.

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References

  1. Betea D., Elliptically distributed lozenge tilings of a hexagon, arXiv:1110.4176.
  2. Borodin A., Ferrari P.L., Prähofer M., Sasamoto T., Warren J., Maximum of Dyson Brownian motion and non-colliding systems with a boundary, Electron. Commun. Probab. 14 (2009), 486-494, arXiv:0905.3989.
  3. Borodin A., Gorin V., Rains E.M., $q$-distributions on boxed plane partitions, Selecta Math. (N.S.) 16 (2010), 731-789, arXiv:0905.0679.
  4. Borodin A., Rains E.M., Eynard-Mehta theorem, Schur process, and their Pfaffian analogs, J. Stat. Phys. 121 (2005), 291-317, math-ph/0409059.
  5. Borodin A.N., Salminen P., Handbook of Brownian motion - facts and formulae, 2nd ed., Probability and its Applications, Birkhäuser Verlag, Basel, 2002.
  6. Dyson F.J., A Brownian-motion model for the eigenvalues of a random matrix, J. Math. Phys. 3 (1962), 1191-1198.
  7. Felder G., Varchenko A., Three formulae for eigenfunctions of integrable Schrödinger operators, Compositio Math. 107 (1997), 143-175, hep-th/9511120.
  8. Forrester P.J., Log-gases and random matrices, London Mathematical Society Monographs Series, Vol. 34, Princeton University Press, Princeton, NJ, 2010.
  9. Gessel I., Viennot G., Binomial determinants, paths, and hook length formulae, Adv. Math. 58 (1985), 300-321.
  10. Ito M., Noumi M., A generalization of the Sears-Slater transformation and elliptic Lagrange interpolation of type $BC_n$, Adv. Math. 299 (2016), 361-380, arXiv:1506.07267.
  11. Ito M., Noumi M., Derivation of a $BC_n$ elliptic summation formula via the fundamental invariants, Constr. Approx. 45 (2017), 33-46, arXiv:1504.07108.
  12. Ito M., Noumi M., Evaluation of the $BC_n$ elliptic Selberg integral via the fundamental invariants, Proc. Amer. Math. Soc. 145 (2017), 689-703, arXiv:1504.07317.
  13. Karlin S., McGregor J., Coincidence probabilities, Pacific J. Math. 9 (1959), 1141-1164.
  14. Katori M., Determinantal martingales and noncolliding diffusion processes, Stochastic Process. Appl. 124 (2014), 3724-3768, arXiv:1305.4412.
  15. Katori M., Bessel processes, Schramm-Loewner evolution, and the Dyson model, Springer Briefs in Mathematical Physic, Vol. 11, Springer, Singapore, 2015.
  16. Katori M., Elliptic determinantal process of type A, Probab. Theory Related Fields 162 (2015), 637-677, arXiv:1311.4146.
  17. Katori M., Elliptic Bessel processes and elliptic Dyson models realized as temporally inhomogeneous processes, J. Math. Phys. 57 (2016), 103302, 32 pages, arXiv:1605.02931.
  18. Katori M., Tanemura H., Symmetry of matrix-valued stochastic processes and noncolliding diffusion particle systems, J. Math. Phys. 45 (2004), 3058-3085, math-ph/0402061.
  19. Katori M., Tanemura H., Infinite systems of noncolliding generalized meanders and Riemann-Liouville differintegrals, Probab. Theory Related Fields 138 (2007), 113-156, math.PR/0506187.
  20. Katori M., Tanemura H., Noncolliding Brownian motion and determinantal processes, J. Stat. Phys. 129 (2007), 1233-1277, arXiv:0705.2460.
  21. Katori M., Tanemura H., Zeros of Airy function and relaxation process, J. Stat. Phys. 136 (2009), 1177-1204, arXiv:0906.3666.
  22. Katori M., Tanemura H., Non-equilibrium dynamics of Dyson's model with an infinite number of particles, Comm. Math. Phys. 293 (2010), 469-497, arXiv:0812.4108.
  23. Katori M., Tanemura H., Noncolliding squared Bessel processes, J. Stat. Phys. 142 (2011), 592-615, arXiv:1008.0144.
  24. König W., O'Connell N., Eigenvalues of the Laguerre process as non-colliding squared Bessel processes, Electron. Comm. Probab. 6 (2001), 107-114.
  25. Koprucki T., Wagner H.J., New exact ground states for one-dimensional quantum many-body systems, J. Stat. Phys. 100 (2000), 779-790.
  26. Krattenthaler C., Advanced determinant calculus: a complement, Linear Algebra Appl. 411 (2005), 68-166, math.CO/0503507.
  27. Levin B.Ya., Lectures on entire functions, Translations of Mathematical Monographs, Vol. 150, Amer. Math. Soc., Providence, RI, 1996.
  28. Liechty K., Wang D., Nonintersecting Brownian bridges between reflecting or absorbing walls, Adv. Math. 309 (2017), 155-208, arXiv:1608.08712.
  29. Lindström B., On the vector representations of induced matroids, Bull. London Math. Soc. 5 (1973), 85-90.
  30. Macdonald I.G., Affine root systems and Dedekind's $\eta$-function, Invent. Math. 15 (1972), 91-143.
  31. Olshanetsky M.A., Perelomov A.M., Quantum integrable systems related to Lie algebras, Phys. Rep. 94 (1983), 313-404.
  32. Olver F.W.J., Lozier D.W., Boisvert R.F., Clark C.W. (Editors), NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC, Cambridge University Press, Cambridge, 2010, available at http://dlmf.nist.gov.
  33. Revuz D., Yor M., Continuous martingales and Brownian motion, Grundlehren der Mathematischen Wissenschaften, Vol. 293, 3rd ed., Springer-Verlag, Berlin, 1999.
  34. Rosengren H., Schlosser M., Elliptic determinant evaluations and the Macdonald identities for affine root systems, Compos. Math. 142 (2006), 937-961, math.CA/0505213.
  35. Schlosser M., A noncommutative weight-dependent generalization of the binomial theorem, arXiv:1106.2112.
  36. Schlosser M., Elliptic enumeration of nonintersecting lattice paths, J. Combin. Theory Ser. A 114 (2007), 505-521, math.CO/0602260.
  37. Schlosser M., Yoo M., Weight-dependent commutation relations and combinatorial identities, arXiv:1610.08680.
  38. Spohn H., Interacting Brownian particles: a study of Dyson's model, in Hydrodynamic Behavior and Interacting Particle Systems (Minneapolis, Minn., 1986), IMA Vol. Math. Appl., Vol. 9, Springer, New York, 1987, 151-179.
  39. Sutherland B., One-dimensional plasma as an example of a Wigner solid, Phys. Rev. Lett. 35 (1975), 185-188.
  40. Takemura K., On the eigenstates of the elliptic Calogero-Moser model, Lett. Math. Phys. 53 (2000), 181-194, math.QA/0002104.
  41. Tracy C.A., Widom H., Nonintersecting Brownian excursions, Ann. Appl. Probab. 17 (2007), 953-979, math.PR/0607321.
  42. Warnaar S.O., Summation and transformation formulas for elliptic hypergeometric series, Constr. Approx. 18 (2002), 479-502, math.QA/0001006.
  43. Whittaker E.T., Watson G.N., A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996.

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