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

SIGMA 17 (2021), 021, 34 pages      arXiv:1912.06067

Parameter Permutation Symmetry in Particle Systems and Random Polymers

Leonid Petrov ab
a) University of Virginia, Department of Mathematics, 141 Cabell Drive, Kerchof Hall, P.O. Box 400137, Charlottesville, VA 22904, USA
b) Institute for Information Transmission Problems, Bolshoy Karetny per. 19, Moscow, 127994, Russia

Received October 26, 2020, in final form February 20, 2021; Published online March 06, 2021

Many integrable stochastic particle systems in one space dimension (such as TASEP - totally asymmetric simple exclusion process - and its various deformations, with a notable exception of ASEP) remain integrable when we equip each particle $x_i$ with its own jump rate parameter $\nu_i$. It is a consequence of integrability that the distribution of each particle $x_n(t)$ in a system started from the step initial configuration depends on the parameters $\nu_j$, $j\le n$, in a symmetric way. A transposition $\nu_n \leftrightarrow \nu_{n+1}$ of the parameters thus affects only the distribution of $x_n(t)$. For $q$-Hahn TASEP and its degenerations ($q$-TASEP and directed beta polymer) we realize the transposition $\nu_n \leftrightarrow \nu_{n+1}$ as an explicit Markov swap operator acting on the single particle $x_n(t)$. For beta polymer, the swap operator can be interpreted as a simple modification of the lattice on which the polymer is considered. Our main tools are Markov duality and contour integral formulas for joint moments. In particular, our constructions lead to a continuous time Markov process $\mathsf{Q}^{(\mathsf{t})}$ preserving the time $\mathsf{t}$ distribution of the $q$-TASEP (with step initial configuration, where $\mathsf{t}\in \mathbb{R}_{>0}$ is fixed). The dual system is a certain transient modification of the stochastic $q$-Boson system. We identify asymptotic survival probabilities of this transient process with $q$-moments of the $q$-TASEP, and use this to show the convergence of the process $\mathsf{Q}^{(\mathsf{t})}$ with arbitrary initial data to its stationary distribution. Setting $q=0$, we recover the results about the usual TASEP established recently in [arXiv:1907.09155] by a different approach based on Gibbs ensembles of interlacing particles in two dimensions.

Key words: $q$-TASEP; stochastic $q$-Boson system; stationary distribution; coordinate Bethe ansatz; $q$-Hahn TASEP.

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  1. Assiotis T., Determinantal structures in space-inhomogeneous dynamics on interlacing arrays, Ann. Henri Poincaré 21 (2020), 909-940, arXiv:1910.09500.
  2. Barraquand G., A short proof of a symmetry identity for the $q$-Hahn distribution, Electron. Commun. Probab. 19 (2014), no. 50, 3 pages, arXiv:1404.4265.
  3. Barraquand G., Corwin I., The $q$-Hahn asymmetric exclusion process, Ann. Appl. Probab. 26 (2016), 2304-2356, arXiv:1501.03445.
  4. Barraquand G., Corwin I., Random-walk in beta-distributed random environment, Probab. Theory Related Fields 167 (2017), 1057-1116, arXiv:1503.04117.
  5. Basu R., Sarkar S., Sly A., Last passage percolation with a defect line and the solution of the slow bond problem, arXiv:1408.3464.
  6. Basu R., Sarkar S., Sly A., Invariant measures for TASEP with a slow bond, arXiv:1704.07799.
  7. Borodin A., Corwin I., Macdonald processes, Probab. Theory Related Fields 158 (2014), 225-400, arXiv:1111.4408.
  8. Borodin A., Corwin I., Gorin V., Stochastic six-vertex model, Duke Math. J. 165 (2016), 563-624, arXiv:1407.6729.
  9. Borodin A., Corwin I., Petrov L., Sasamoto T., Spectral theory for interacting particle systems solvable by coordinate Bethe ansatz, Comm. Math. Phys. 339 (2015), 1167-1245, arXiv:1407.8534.
  10. Borodin A., Corwin I., Sasamoto T., From duality to determinants for $q$-TASEP and ASEP, Ann. Probab. 42 (2014), 2314-2382, arXiv:1207.5035.
  11. Borodin A., Ferrari P.L., Anisotropic growth of random surfaces in $2+1$ dimensions, Comm. Math. Phys. 325 (2014), 603-684, arXiv:0804.3035.
  12. Borodin A., Gorin V., Wheeler M., Shift-invariance for vertex models and polymers, arXiv:1912.02957.
  13. Borodin A., Petrov L., Integrable probability: stochastic vertex models and symmetric functions, in Stochastic Processes and Random Matrices, Oxford University Press, Oxford, 2017, 26-131, arXiv:1605.01349.
  14. Borodin A., Petrov L., Higher spin six vertex model and symmetric rational functions, Selecta Math. (N.S.) 24 (2018), 751-874, arXiv:1601.05770.
  15. Borodin A., Petrov L., Inhomogeneous exponential jump model, Probab. Theory Related Fields 172 (2018), 323-385, arXiv:1703.03857.
  16. Borodin A., Wheeler M., Coloured stochastic vertex models and their spectral theory, arXiv:1808.01866.
  17. Bufetov A., Mucciconi M., Petrov L., Yang-Baxter random fields and stochastic vertex models, Adv. Math., to appear, arXiv:1905.06815.
  18. Corwin I., Invariance of polymer partition functions under the geometric RSK correspondence, arXiv:2001.01867.
  19. Corwin I., The Kardar-Parisi-Zhang equation and universality class, Random Matrices Theory Appl. 1 (2012), 1130001, 76 pages, arXiv:1106.1596.
  20. Corwin I., The $q$-Hahn boson process and $q$-Hahn TASEP, Int. Math. Res. Not. 2015 (2015), 5577-5603, arXiv:1401.3321.
  21. Corwin I., Kardar-Parisi-Zhang universality, Notices Amer. Math. Soc. 63 (2016), 230-239.
  22. Corwin I., Matveev K., Petrov L., The $q$-Hahn PushTASEP, Int. Math. Res. Not. 2021 (2021), 2210-2249, arXiv:1811.06475.
  23. Corwin I., O'Connell N., Seppäläinen T., Zygouras N., Tropical combinatorics and Whittaker functions, Duke Math. J. 163 (2014), 513-563, arXiv:1110.3489.
  24. Corwin I., Petrov L., Stochastic higher spin vertex models on the line, Comm. Math. Phys. 343 (2016), 651-700, arXiv:1502.07374.
  25. Costin O., Lebowitz J.L., Speer E.R., Troiani A., The blockage problem, Bull. Inst. Math. Acad. Sin. (N.S.) 8 (2013), 49-72, arXiv:1207.6555.
  26. Dauvergne D., Hidden invariance of last passage percolation and directed polymers, arXiv:2002.09459.
  27. Ferrari P.L., VetHo B., Tracy-Widom asymptotics for $q$-TASEP, Ann. Inst. Henri Poincaré Probab. Stat. 51 (2015), 1465-1485, arXiv:1310.2515.
  28. Galashin P., Symmetries of stochastic colored vertex models, arXiv:2003.06330.
  29. Gravner J., Tracy C.A., Widom H., Fluctuations in the composite regime of a disordered growth model, Comm. Math. Phys. 229 (2002), 433-458, arXiv:math.PR/0111036.
  30. Gwa L.H., Spohn H., Six-vertex model, roughened surfaces, and an asymmetric spin Hamiltonian, Phys. Rev. Lett. 68 (1992), 725-728.
  31. Halpin-Healy T., Takeuchi K.A., A KPZ cocktail - shaken, not stirred ... toasting 30 years of kinetically roughened surfaces, J. Stat. Phys. 160 (2015), 794-814, arXiv:1505.01910.
  32. Its A.R., Tracy C.A., Widom H., Random words, Toeplitz determinants and integrable systems. II, Phys. D 152/153 (2001), 199-224, arXiv:nlin.SI/0004018.
  33. Janowsky S.A., Lebowitz J.L., Finite-size effects and shock fluctuations in the asymmetric simple-exclusion process, Phys. Rev. A 45 (1992), 618-625.
  34. Johansson K., Shape fluctuations and random matrices, Comm. Math. Phys. 209 (2000), 437-476, arXiv:math.CO/9903134.
  35. Knizel A., Petrov L., Saenz A., Generalizations of TASEP in discrete and continuous inhomogeneous space, Comm. Math. Phys. 372 (2019), 797-864, arXiv:1808.09855.
  36. Mucciconi M., Petrov L., Spin $q$-Whittaker polynomials and deformed quantum Toda, arXiv:2003.14260.
  37. O'Connell N., Directed polymers and the quantum Toda lattice, Ann. Probab. 40 (2012), 437-458, arXiv:0910.0069.
  38. O'Connell N., Seppäläinen T., Zygouras N., Geometric RSK correspondence, Whittaker functions and symmetrized random polymers, Invent. Math. 197 (2014), 361-416, arXiv:1110.3489.
  39. Okounkov A., Reshetikhin N., Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram, J. Amer. Math. Soc. 16 (2003), 581-603, arXiv:math.CO/0107056.
  40. Petrov L., PushTASEP in inhomogeneous space, Electron. J. Probab. 25 (2020), 114, 25 pages, arXiv:1910.08994.
  41. Petrov L., Saenz A., Mapping TASEP back in time, arXiv:1907.09155.
  42. Povolotsky A.M., On the integrability of zero-range chipping models with factorized steady states, J. Phys. A: Math. Theor. 46 (2013), 465205, 25 pages, arXiv:1308.3250.
  43. Quastel J., Spohn H., The one-dimensional KPZ equation and its universality class, J. Stat. Phys. 160 (2015), 965-984, arXiv:1503.06185.
  44. Rákos A., Schütz G.M., Bethe ansatz and current distribution for the TASEP with particle-dependent hopping rates, Markov Process. Related Fields 12 (2006), 323-334, arXiv:cond-mat/0506525.
  45. Sasamoto T., Wadati M., Exact results for one-dimensional totally asymmetric diffusion models, J. Phys. A: Math. Gen. 31 (1998), 6057-6071.
  46. Seppäläinen T., Hydrodynamic profiles for the totally asymmetric exclusion process with a slow bond, J. Stat. Phys. 102 (2001), 69-96, arXiv:math.PR/0003049.
  47. Seppäläinen T., Scaling for a one-dimensional directed polymer with boundary conditions, Ann. Probab. 40 (2012), 19-73, arXiv:0911.2446.
  48. Takeyama Y., A deformation of affine Hecke algebra and integrable stochastic particle system, J. Phys. A: Math. Theor. 47 (2014), 465203, 19 pages, arXiv:1407.1960.
  49. Thiery T., Le Doussal P., On integrable directed polymer models on the square lattice, J. Phys. A: Math. Theor. 48 (2015), 465001, 41 pages, arXiv:1506.05006.
  50. Tracy C.A., Widom H., Integral formulas for the asymmetric simple exclusion process, Comm. Math. Phys. 279 (2008), 815-844, Erratum, Comm. Math. Phys. 304 (2011), 875-878, arXiv:0704.2633.
  51. Tracy C.A., Widom H., Asymptotics in ASEP with step initial condition, Comm. Math. Phys. 290 (2009), 129-154, arXiv:0807.1713.
  52. Vető B., Tracy-Widom limit of $q$-Hahn TASEP, Electron. J. Probab. 20 (2015), 102, 22 pages, arXiv:1407.2787.

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