Parameter Permutation Symmetry in Particle Systems and Random Polymers

Leonid Petrov

Symmetry, integrability and geometry: methods and applications, Tome 17 (2021)

Résumé

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.

Détail

Keywords: stochastic $q$-Boson system, stationary distribution, coordinate Bethe ansatz
Mots-clés : $q$-TASEP, $q$-Hahn TASEP.

Bibliographie

[1] Assiotis T., “Determinantal structures in space-inhomogeneous dynamics on interlacing arrays”, Ann. Henri Poincaré, 21 (2020), 909–940, arXiv: 1910.09500 | DOI | MR | Zbl

[2] Barraquand G., “A short proof of a symmetry identity for the $q$-Hahn distribution”, Electron. Commun. Probab., 19:50 (2014), 3 pp., arXiv: 1404.4265 | DOI | MR | Zbl

[3] Barraquand G., Corwin I., “The $q$-Hahn asymmetric exclusion process”, Ann. Appl. Probab., 26 (2016), 2304–2356, arXiv: 1501.03445 | DOI | MR | Zbl

[4] Barraquand G., Corwin I., “Random-walk in beta-distributed random environment”, Probab. Theory Related Fields, 167 (2017), 1057–1116, arXiv: 1503.04117 | DOI | MR | Zbl

[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 | MR

[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 | DOI | MR | Zbl

[8] Borodin A., Corwin I., Gorin V., “Stochastic six-vertex model”, Duke Math. J., 165 (2016), 563–624, arXiv: 1407.6729 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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”, Stochastic Processes and Random Matrices, Oxford University Press, Oxford, 2017, 26–131, arXiv: 1605.01349 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[15] Borodin A., Petrov L., “Inhomogeneous exponential jump model”, Probab. Theory Related Fields, 172 (2018), 323–385, arXiv: 1703.03857 | DOI | MR | Zbl

[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 | MR

[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 pp., arXiv: 1106.1596 | DOI | MR | Zbl

[20] Corwin I., “The $q$-Hahn boson process and $q$-Hahn TASEP”, Int. Math. Res. Not., 2015 (2015), 5577–5603, arXiv: 1401.3321 | DOI | MR | Zbl

[21] Corwin I., “Kardar–Parisi–Zhang universality”, Notices Amer. Math. Soc., 63 (2016), 230–239 | DOI | MR | Zbl

[22] Corwin I., Matveev K., Petrov L., “The $q$-Hahn PushTASEP”, Int. Math. Res. Not., 2021 (2021), 2210–2249, arXiv: 1811.06475 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[24] Corwin I., Petrov L., “Stochastic higher spin vertex models on the line”, Comm. Math. Phys., 343 (2016), 651–700, arXiv: 1502.07374 | DOI | MR | Zbl

[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 | MR | Zbl

[26] Dauvergne D., Hidden invariance of last passage percolation and directed polymers, arXiv: 2002.09459

[27] Ferrari P. L., Vető B., “Tracy–Widom asymptotics for $q$-TASEP”, Ann. Inst. Henri Poincaré Probab. Stat., 51 (2015), 1465–1485, arXiv: 1310.2515 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[30] Gwa L. H., Spohn H., “Six-vertex model, roughened surfaces, and an asymmetric spin Hamiltonian”, Phys. Rev. Lett., 68 (1992), 725–728 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR

[34] Johansson K., “Shape fluctuations and random matrices”, Comm. Math. Phys., 209 (2000), 437–476, arXiv: math.CO/9903134 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[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 | DOI | MR

[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 | DOI | MR

[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 | DOI | MR | Zbl

[40] Petrov L., “PushTASEP in inhomogeneous space”, Electron. J. Probab., 25 (2020), 114, 25 pp., arXiv: 1910.08994 | DOI | MR | Zbl

[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 pp., arXiv: 1308.3250 | DOI | MR | Zbl

[43] Quastel J., Spohn H., “The one-dimensional KPZ equation and its universality class”, J. Stat. Phys., 160 (2015), 965–984, arXiv: 1503.06185 | DOI | MR | Zbl

[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 | MR | Zbl

[45] Sasamoto T., Wadati M., “Exact results for one-dimensional totally asymmetric diffusion models”, J. Phys. A: Math. Gen., 31 (1998), 6057–6071 | DOI | MR | Zbl

[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 | DOI | Zbl

[47] Seppäläinen T., “Scaling for a one-dimensional directed polymer with boundary conditions”, Ann. Probab., 40 (2012), 19–73, arXiv: 0911.2446 | DOI | MR | Zbl

[48] Takeyama Y., “A deformation of affine Hecke algebra and integrable stochastic particle system”, J. Phys. A: Math. Theor., 47 (2014), 465203, 19 pp., arXiv: 1407.1960 | DOI | MR | Zbl

[49] Thiery T., Le Doussal P., “On integrable directed polymer models on the square lattice”, J. Phys. A: Math. Theor., 48 (2015), 465001, 41 pp., arXiv: 1506.05006 | DOI | MR | Zbl

[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 | DOI | MR | Zbl | DOI | MR | Zbl

[51] Tracy C. A., Widom H., “Asymptotics in ASEP with step initial condition”, Comm. Math. Phys., 290 (2009), 129–154, arXiv: 0807.1713 | DOI | MR | Zbl

[52] Vető B., “Tracy-Widom limit of $q$-Hahn TASEP”, Electron. J. Probab., 20 (2015), 102, 22 pp., arXiv: 1407.2787 | DOI | MR | Zbl