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

SIGMA 15 (2019), 053, 27 pages      arXiv:1812.08553

Orthogonal Dualities of Markov Processes and Unitary Symmetries

Gioia Carinci a, Chiara Franceschini b, Cristian Giardinà c, Wolter Groenevelt a and Frank Redig a
a) Technische Universiteit Delft, DIAM, P.O. Box 5031, 2600 GA Delft, The Netherlands
b) Center for Mathematical Analysis Geometry and Dynamical Systems, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
c) University of Modena and Reggio Emilia, FIM, via G. Campi 213/b, 41125 Modena, Italy

Received December 24, 2018, in final form July 05, 2019; Published online July 12, 2019

We study self-duality for interacting particle systems, where the particles move as continuous time random walkers having either exclusion interaction or inclusion interaction. We show that orthogonal self-dualities arise from unitary symmetries of the Markov generator. For these symmetries we provide two equivalent expressions that are related by the Baker-Campbell-Hausdorff formula. The first expression is the exponential of an anti Hermitian operator and thus is unitary by inspection; the second expression is factorized into three terms and is proved to be unitary by using generating functions. The factorized form is also obtained by using an independent approach based on scalar products, which is a new method of independent interest that we introduce to derive (bi)orthogonal duality functions from non-orthogonal duality functions.

Key words: stochastic duality; interacting particle systems; Lie algebras; orthogonal polynomials.

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  1. Ayala M., Carinci G., Redig F., Quantitative Boltzmann-Gibbs principles via orthogonal polynomial duality, J. Stat. Phys. 171 (2018), 980-999, arXiv:1712.08492.
  2. Borodin A., Corwin I., Gorin V., Stochastic six-vertex model, Duke Math. J. 165 (2016), 563-624, arXiv:1407.6729.
  3. Borodin A., Corwin I., Sasamoto T., From duality to determinants for $q$-TASEP and ASEP, Ann. Probab. 42 (2014), 2314-2382, arXiv:1207.5035.
  4. Carinci G., Giardinà C., Giberti C., Redig F., Duality for stochastic models of transport, J. Stat. Phys. 152 (2013), 657-697, arXiv:1212.3154.
  5. Carinci G., Giardinà C., Giberti C., Redig F., Dualities in population genetics: a fresh look with new dualities, Stochastic Process. Appl. 125 (2015), 941-969, arXiv:1302.3206.
  6. Carinci G., Giardinà C., Redig F., Sasamoto T., Asymmetric stochastic transport models with $\mathcal{U}_q(\mathfrak{su}(1,1))$ symmetry, J. Stat. Phys. 163 (2016), 239-279, arXiv:1507.01478.
  7. Carinci G., Giardinà C., Redig F., Sasamoto T., A generalized asymmetric exclusion process with $U_q(\mathfrak{sl}_2)$ stochastic duality, Probab. Theory Related Fields 166 (2016), 887-933, arXiv:1407.3367.
  8. Corwin I., Ghosal P., Shen H., Tsai L.-C., Stochastic PDE limit of the six vertex model, arXiv:1803.08120.
  9. Corwin I., Petrov L., Stochastic higher spin vertex models on the line, Comm. Math. Phys. 343 (2016), 651-700, arXiv:1502.07374.
  10. Corwin I., Shen H., Tsai L.-C., ${\rm ASEP}(q,j)$ converges to the KPZ equation, Ann. Inst. Henri Poincar'e Probab. Stat. 54 (2018), 995-1012, arXiv:1602.01908.
  11. De Masi A., Presutti E., Mathematical methods for hydrodynamic limits, Lecture Notes in Mathematics, Vol. 1501, Springer-Verlag, Berlin, 1991.
  12. Franceschini C., Giardinà C., Stochastic duality and orthogonal polynomials, arXiv:1701.09115.
  13. Franceschini C., Giardinà C., Groenevelt W., Self-duality of Markov processes and intertwining functions, Math. Phys. Anal. Geom. 21 (2018), Art. 29, 21 pages, arXiv:1801.09433.
  14. Giardinà C., Kurchan J., Redig F., Duality and exact correlations for a model of heat conduction, J. Math. Phys. 48 (2007), 033301, 15 pages, arXiv:cond-mat/0612198.
  15. Giardinà C., Kurchan J., Redig F., Vafayi K., Duality and hidden symmetries in interacting particle systems, J. Stat. Phys. 135 (2009), 25-55, arXiv:0810.1202.
  16. Giardinà C., Redig F., Vafayi K., Correlation inequalities for interacting particle systems with duality, J. Stat. Phys. 141 (2010), 242-263, arXiv:0906.4664.
  17. Groenevelt W., Orthogonal stochastic duality functions from Lie algebra representations, J. Stat. Phys. 174 (2019), 97-119, arXiv:1709.05997.
  18. Imamura T., Sasamoto T., Current moments of 1D ASEP by duality, J. Stat. Phys. 142 (2011), 919-930, arXiv:1011.4588.
  19. Jansen S., Kurt N., On the notion(s) of duality for Markov processes, Probab. Surv. 11 (2014), 59-120, arXiv:1210.7193.
  20. Kipnis C., Marchioro C., Presutti E., Heat flow in an exactly solvable model, J. Stat. Phys. 27 (1982), 65-74.
  21. Koekoek R., Lesky P.A., Swarttouw R.F., Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
  22. Kuan J., An algebraic construction of duality functions for the stochastic $\mathcal{U}_q\big( A_n^{(1)}\big)$ vertex model and its degenerations, Comm. Math. Phys. 359 (2018), 121-187, arXiv:1701.04468.
  23. Liggett T.M., Interacting particle systems, Classics in Mathematics, Springer-Verlag, Berlin, 2005.
  24. Möhle M., The concept of duality and applications to Markov processes arising in neutral population genetics models, Bernoulli 5 (1999), 761-777.
  25. Ohkubo J., On dualities for SSEP and ASEP with open boundary conditions, J. Phys. A: Math. Theor. 50 (2017), 095004, 21 pages, arXiv:1606.05447.
  26. Redig F., Sau F., Factorized duality, stationary product measures and generating functions, J. Stat. Phys. 172 (2018), 980-1008, arXiv:1702.07237.
  27. Rosengren H., A new quantum algebraic interpretation of the Askey-Wilson polynomials, in $q$-Series from a Contemporary Perspective (South Hadley, MA, 1998), Contemp. Math., Vol. 254, Amer. Math. Soc., Providence, RI, 2000, 371-394.
  28. Schütz G.M., Duality relations for asymmetric exclusion processes, J. Stat. Phys. 86 (1997), 1265-1287.
  29. Schütz G.M., Sandow S., Non-Abelian symmetries of stochastic processes: Derivation of correlation functions for random-vertex models and disordered-interacting-particle systems, Phys. Rev. E 49 (1994), 2726-2741.
  30. Spitzer F., Interaction of Markov processes, Adv. Math. 5 (1970), 246-290.
  31. Spohn H., Long range correlations for stochastic lattice gases in a nonequilibrium steady state, J. Phys. A: Math. Gen. 16 (1983), 4275-4291.
  32. Truax D.R., Baker-Campbell-Hausdorff relations and unitarity of ${\rm SU}(2)$ and ${\rm SU}(1,1)$ squeeze operators, Phys. Rev. D 31 (1985), 1988-1991.

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