Abstract
A recent paper (Kuniba in Nucl Phys B 913:248–277, 2016) introduced the stochastic \({\mathcal{U}_q(A_n^{(1)})}\) vertex model. The stochastic S-matrix is related to the R-matrix of the quantum group \({\mathcal{U}_q(A_n^{(1)})}\) by a gauge transformation. We will show that a certain function \({D^+_{\vec{m}}}\) intertwines with the transfer matrix and its space reversal. When interpreting the transfer matrix as the transition matrix of a discrete-time totally asymmetric particle system on the one-dimensional lattice \({\mathbb{Z}}\), the function \({D^+_{\vec{m}}}\) becomes a Markov duality function \({D_{\vec{m}}}\) which only depends on q and the vertical spin parameters \({\{\mu_x\}}\). By considering degenerations in the spectral parameter, the duality results also hold on a finite lattice with closed boundary conditions, and for a continuous-time degeneration. This duality function had previously appeared in a multi-species ASEP(q, j) process (Kuan in A multi-species ASEP(q, j) and q-TAZRP with stochastic duality, 2017). The proof here uses that the R-matrix intertwines with the co-product, but does not explicitly use the Yang–Baxter equation. It will also be shown that the stochastic \({\mathcal{U}_q(A_n^{(1)})}\) is a multi-species version of a stochastic vertex model studied in Borodin and Petrov (Higher spin six vertex model and symmetric rational functions, 2016) and Corwin and Petrov (Commun Math Phys 343:651–700, 2016). This will be done by generalizing the fusion process of Corwin and Petrov (2016) and showing that it matches the fusion of Kulish and yu (Lett Math Phys 5:393–403, 1981) up to the gauge transformation. We also show, by direct computation, that the multi-species q-Hahn Boson process (which arises at a special value of the spectral parameter) also satisfies duality with respect to \({D_{\infty}}\), generalizing the single-species result of Corwin (Int Math Res Not 2015:5577–5603, 2015).
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References
Alcaraz F.C., Rittenberg V.: Reaction–diffusion processes as physical realizations of Hecke algebras. Phys. Lett. B 314(3), 377–380 (1993)
Andrews, G.E., Askey, R., Roy, R.: Special Functions. vol. 71. Cambridge University Press, Cambridge (2001)
Barraquand G.: A short proof of a symmetry identity for the q–Hahn distribution. Electron. Commun. Probab. 19(50), 1–3 (2014) https://doi.org/10.1214/ECP.v19-3674
Barraquand G., orwin I.: The q-Hahn asymmetric exclusion process. Ann. Appl. Probab. 26(4), 2304–2356 (2016)
Belitsky V., Schütz G.M.: Self-duality for the two-component asymmetric simple exclusion process. J. Math. Phys. 56, 083302 (2015) https://doi.org/10.1063/1.4929663
Belitsky V., Schütz G.M.: Quantum algebra symmetry and reversible measures for the ASEP with second-class particles. J. Stat. Phys. 161(4), 821–842 (2015) https://doi.org/10.1007/s10955-015-1363-1
Belitsky, V., Schütz, G.M.: Self-duality and shock dynamics in the n-component priority ASEP. (2016). arXiv:1606.04587v1
Borodin, A., Corwin, I.: Discrete time q–TASEPs. Int. Math. Res. Not. (2013). https://dx.doi.org/10.1093/imrn/rnt206
Borodin A., Corwin I., Gorin V.: Stochastic six-vertex model. Duke Math. J. 165(3), 563–624 (2016) https://doi.org/10.1215/00127094-3166843
Borodin A., Corwin I., Sasamoto T.: From duality to determinants for q–TASEP and ASEP. Ann. Probab. 42(6), 2314–2382 (2014) https://doi.org/10.1214/13-AOP868
Borodin A., Corwin I., Petrov L., Sasamoto T.: Spectral theory for the q-Boson particle system. Compos. Math. 151, 1–67 (2015)
Borodin A., Petrov L.: Higher spin six vertex model and symmetric rational functions. Sel. Math. Newser. 1, 1–24 (2016)
Borodin, A., Petrov, L.: Lectures on Integrable probability: stochastic vertex models and symmetric functions. (2016). arXiv:1605.01349v1
Bosnjak G., Mangazeev V.: Construction of R-matrices for symmetric tensor representations related to \({U_q(\widehat{sl_n})}\). J. Phys. A Math. Theor. 49, 495204 (2016) https://doi.org/10.1088/1751-8113/49/49/495204
Cantini, L.: Asymmetric Simple Exclusion Process with open boundaries and Koornwinder polynomials. J. Phys. A Math. Gen. 37(18) (2004). arXiv:1506.00284v1
Cantini, L., Garbali, A., Gier, J.D., Wheeler, M.: Koornwinder polynomials and the stationary multi-species asymmetric exclusion process with open boundaries. J. Phys. A Math. Theor. 49, 444002 (2016). https://doi.org/10.1088/1751-8113/49/44/444002
Cantini, L., de Gier, J., Wheeler, M.: Matrix product and sum rule for Macdonald polynomials. (2016). arXiv:1602.04392v1
Carinci, G., Giardinà, C., Redig, F., Sasamoto, T.: A generalized asymmetric exclusion process with \({\mathcal{U}_q(\mathfrak{sl}_2)}\) stochastic duality. Probab. Theory Relat. Fields. (2015). https://doi.org/10.1007/s0040-015-0674-0
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(2), 239–279 (2016) https://doi.org/10.1007/s10955-016-1473-4
Corteel S., Mandelshtam O., Williams L.: Combinatorics of the two-species ASEP and Koornwinder moments. Adv. Math. 321, 160–204 (2017) arXiv:1510.05023
Corwin I.: The q-Hahn Boson Process and q-Hahn TASEP. Int. Math. Res. Not. 2015(14), 5577–5603 (2015) https://doi.org/10.1093/imrn/rnu094
Corwin I., Petrov L.: Stochastic higher spin vertex models on the line. Commun. Math. Phys. 343(2), 651–700 (2016) https://doi.org/10.1007/s00220-015-2479-5
Corwin, I., Shen, H., Tsai, L.-C.: ASEP(q,j) converges to the KPZ equation. (2016). arXiv:1602.01908v1
Crampe, N., Finn, C., Ragoucy, E., Vanicat, M.: Integrable boundary conditions for multi-species ASEP. J. Phys. A Math. Theor. (2016). https://doi.org/10.1088/1751-8113/49/37/375201
Fuchs J.: Affine Lie Algebras and Quantum Groups. Cambridge University Press, Cambridge (1995)
Giardinà C., Kurchan J., Redig F., Vafayi K.: Duality and hidden symmetries in interacting particle systems. J. Stat. Phys. 135, 25–55 (2009) https://doi.org/10.1007/s10955-009-9716-2
Giardinà C., Redig F., Vafayi K.: Correlation inequalities for interacting particle systems with duality. J. Stat. Phys. 141, 242–263 (2010) https://doi.org/10.1007/s10955-010-0055-0
Gwa L–H., Spohn H.: Bethe solution for the dynamical-scaling exponent of the noisy Burgers equation. Phys. Rev. A. 46, 844–854 (1992)
Imamura T., Sasamoto T.: Current moments of 1D ASEP by duality. J. Stat. Phys. 142(5), 919–930 (2011) https://doi.org/10.1007/s10955-011-0149-3
Jansen S., Kurt N.: On the notion(s) of duality for Markov processes. Probab. Surv. 11, 59–120 (2014)
Karimipour, V.: A multi-species asymmetric exclusion process, steady state and correlation functions on a periodic lattice. EPL (Europhys. Lett.) (1999). https://doi.org/10.1209/epl/i1999-00389-2
Kemeny J.G., Snell J.L.: Finite Markov Chains. Springer, Berlin (1976)
Kuan J.: Stochastic duality of ASEP with two particle types via symmetry of quantum groups of rank two. J. Phys A. 49(11), 29 (2016) https://doi.org/10.1088/1751-8113/49/11/115002
Kuan, J.: A multi-species ASEP(q,j) and q-TAZRP with stochastic duality (2017). arXiv:1605.00691v1 (to appear in Int. Mat. Res. Not.)
Kulish P.P., Reshetikhin N.Yu., Sklyanin E.K.: Yang–Baxter equation and representation theory: I. Lett. Math. Phys. 5, 393–403 (1981)
Kuniba, K., Mangazeev, V., Maruyama, M.: Stochastic R Matrix for U q (A (1) n ). Nucl. Phys. B 913, 248–277 (2016). arXiv:1604.08304v4
Kuniba, A., Okado, M.: Matrix product formula for \({\mathcal{U}_q(A_2^{(1)})}\)-zero range process. J. Phys. A Math. Theor. 50, 4 (2016). arXiv:1608.02779v1
Kuniba A., Okado M.: A q-boson representation of Zamolodchikov–Faddeev algebra for stochastic R matrix of U q (A (1) n ). Lett. Math. Phys. 107, 1111 (2017) arXiv:1610.00531v1
Kuniba A., Okado M., Sergeev S.: Tetrahedron equation and generalized quantum groups. J. Phys. A Math. Theor. 48, 304001 (2015) https://doi.org/10.1088/1751-8113/48/30/304001
Liggett T.M.: Coupling the simple exclusion process. Ann. Probab. 4, 339–356 (1976) https://doi.org/10.1214/aop/1176996084
Liggett T.M.: Interacting Particle Systems. Springer, Berlin (2005)
Lusztig G.: Introduction to Quantum Groups. Birkhäuser, Boston (1993)
Mandelshtam, O.: Matrix ansatz and combinatorics of the k-species PASEP. (2015). arXiv:1508.04115v1
Mandelshtam, O., Viennot, X.: Tableaux combinatorics of the two-species PASEP. (2015). arXiv:1506.01980v1
Ohkubo, J.: On dualities for SSEP and ASEP with open boundary conditions J. Phys. A Math. Theor. 50 095004 (2017). arXiv:1606.05447v1
Pitman J.W., Rogers L.C.G.: Markov functions. Ann. Probab. 9(4), 573–582 (1981)
Povolotsky, A.M.: On the integrability of zero-range chipping models with factorized steady states. J. Phys. A Math. Theor. (2013). https://doi.org/10.1088/1751-8113/46/46/465205
Povolotsky A.M., Priezzhev V.B.: Determinant solution for the totally asymmetric exclusion process with parallel update. J. Stat. Mech. 07, P07002 (2006) https://doi.org/10.1088/1742-5468/2006/07/P07002
Prolhac S., Evans M.R., Mallick K.: Matrix product solution of the multispecies partially asymmetric exclusion process. J. Phys. A Math. Theor. 42, 165004 (2009) https://doi.org/10.1088/1751-8113/42/16/165004
Reshetikhin, N.: Lectures on the integrability of the 6-vertex model. (2010). arXiv:1010.5031v1
Sasamoto T., Wadati M.: Exact results for one-dimensional totally asymmetric diffusion models. J. Phys. A. 31(28), 6057–6071 (1998) https://doi.org/10.1088/0305-4470/31/28/019
Schütz G.: Duality relations for asymmetric exclusion processes. J. Stat. Phys. 86(5/6), 1265–1287 (1997) https://doi.org/10.1007/BF02183623
Schütz, G.: Duality relations for the periodic ASEP conditioned on a low current, to appear in from particle systems to partial differential equations III. In: Springer Proceedings in Mathematics and Statistics, vol. 162 (2016). arXiv:1508.03158v1
Schütz G., 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, 2726 (1994) https://doi.org/10.1103/PhysRevE.49.2726
Spitzer F.: Interaction of Markov processes. Adv. Math 5, 246–290 (1970) https://doi.org/10.1016/0001-8708(70)90034-4
Takeyama Y., A deformation of affine Hecke algebra and integrable stochastic particle system. J. Phys. A. (2014). https://doi.org/10.1088/1751-8113/47/46/465203
Takeyama, Y.: Algebraic construction of multi-species q-Boson system. (2015). arXiv:1507.02033
Uchiyama M.: Two-species asymmetric simple exclusion process with open boundaries. Chaos Solitons Fractals 35(2), 398–407 (2008) https://doi.org/10.1016/j.chaos.2006.05.013
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Kuan, J. An Algebraic Construction of Duality Functions for the Stochastic \({\mathcal{U}_q( A_n^{(1)})}\) Vertex Model and Its Degenerations. Commun. Math. Phys. 359, 121–187 (2018). https://doi.org/10.1007/s00220-018-3108-x
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DOI: https://doi.org/10.1007/s00220-018-3108-x