Mapping TASEP Back in time. (English) Zbl 1490.60281
Summary: We obtain a new relation between the distributions \(\mu_t\) at different times \(t\ge 0\) of the continuous-time totally asymmetric simple exclusion process (TASEP) started from the step initial configuration. Namely, we present a continuous-time Markov process with local interactions and particle-dependent rates which maps the TASEP distributions \(\mu_t\) backwards in time. Under the backwards process, particles jump to the left, and the dynamics can be viewed as a version of the discrete-space Hammersley process. Combined with the forward TASEP evolution, this leads to a stationary Markov dynamics preserving \(\mu_t\) which in turn brings new identities for expectations with respect to \(\mu_t\). The construction of the backwards dynamics is based on Markov maps interchanging parameters of Schur processes, and is motivated by bijectivizations of the Yang-Baxter equation. We also present a number of corollaries, extensions, and open questions arising from our constructions.
MSC:
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 05E05 | Symmetric functions and generalizations |
| 60J60 | Diffusion processes |
| 82C22 | Interacting particle systems in time-dependent statistical mechanics |
| 82B23 | Exactly solvable models; Bethe ansatz |
Keywords:
totally asymmetric simple exclusion process; Hammersley process; Schur processes; stationary dynamicsReferences:
| [1] | Aggarwal, A.; Borodin, A.; Bufetov, A., Stochasticization of solutions to the Yang-Baxter equation, Ann. Henri Poincare, 20, 8, 2495-2554 (2019) · Zbl 1439.81057 |
| [2] | Aldous, D.; Diaconis, P., Hammersley’s interacting particle process and longest increasing subsequences, Probab. Theory Relat. Fields, 103, 2, 199-213 (1995) · Zbl 0836.60107 |
| [3] | Andjel, E.; Guiol, H., Long-range exclusion processes, generator and invariant measures, Ann. Probab., 33, 6, 2314-2354 (2005) · Zbl 1099.60064 |
| [4] | Baik, J., Liu, Z.: Multipoint distribution of periodic TASEP. J. AMS (2019). arXiv:1710.03284 [math.PR] · Zbl 1484.60104 |
| [5] | Balazs, M.; Bowen, R., Product blocking measures and a particle system proof of the Jacobi triple product, Ann. Inst. Henri Poincaré B, 54, 1, 514-528 (2018) · Zbl 1396.60100 |
| [6] | Barraquand, G.; Corwin, I., The \(q\)-Hahn asymmetric exclusion process, Ann. Appl. Probab., 26, 4, 2304-2356 (2016) · Zbl 1352.60127 |
| [7] | Basu, R., Ganguly, S.: Time correlation exponents in last passage percolation (2018). arXiv preprint arXiv:1807.09260 [math.PR] · Zbl 1469.60305 |
| [8] | Basu, R., Ganguly, S., Hammond, A.: Fractal geometry of \(Airy_2\) processes coupled via the Airy sheet (2019). arXiv preprint arXiv:1904.01717 [math.PR] · Zbl 1457.82165 |
| [9] | Belitsky, V.; Schütz, G., Self-duality and shock dynamics in the \(n\)-component priority ASEP, Stoch. Process. Appl., 128, 4, 1165-1207 (2018) · Zbl 1391.60229 |
| [10] | Benassi, A.; Fouque, J-P, Hydrodynamical limit for the asymmetric simple exclusion process, Ann. Probab., 15, 2, 546-560 (1987) · Zbl 0623.60120 |
| [11] | Borodin, A., Schur dynamics of the Schur processes, Adv. Math., 228, 4, 2268-2291 (2011) · Zbl 1234.60012 |
| [12] | Borodin, A., On a family of symmetric rational functions, Adv. Math., 306, 973-1018 (2017) · Zbl 1355.05250 |
| [13] | Borodin, A.; Bufetov, A., An irreversible local Markov chain that preserves the six vertex model on a torus, Ann. Inst. Henri Poincaré B, 53, 1, 451-463 (2017) · Zbl 1361.60059 |
| [14] | Borodin, A.; Corwin, I., Macdonald processes, Probab. Theory Relat. Fields, 158, 225-400 (2014) · Zbl 1291.82077 |
| [15] | Borodin, A.; Ferrari, P., Large time asymptotics of growth models on space-like paths I: PushASEP, Electron. J. Probab., 13, 1380-1418 (2008) · Zbl 1187.82084 |
| [16] | Borodin, A.; Ferrari, P., Anisotropic growth of random surfaces in \(2+1\) dimensions, Commun. Math. Phys., 325, 603-684 (2014) · Zbl 1303.82015 |
| [17] | Borodin, A.; Gorin, V., Markov processes of infinitely many nonintersecting random walks, Probab. Theory Relat. Fields, 155, 3-4, 935-997 (2013) · Zbl 1278.60113 |
| [18] | Borodin, A., Gorin, V.: Lectures on integrable probability. In: Probability and Statistical Physics in St. Petersburg, vol. 91. Proceedings of Symposia in Pure Mathematics, pp. 155-214. AMS (2016). arXiv:1212.3351 [math.PR] · Zbl 1388.60157 |
| [19] | Borodin, A.; Gorin, V.; Rains, E., q-Distributions on boxed plane partitions, Sel. Math., 16, 4, 731-789 (2010) · Zbl 1205.82122 |
| [20] | Borodin, A.; Kuan, J., Asymptotics of Plancherel measures for the infinite-dimensional unitary group, Adv. Math., 219, 3, 894-931 (2008) · Zbl 1153.60058 |
| [21] | Borodin, A.; Olshanski, G., Infinite-dimensional diffusions as limits of random walks on partitions, Probab. Theory Relat. Fields, 144, 1, 281-318 (2009) · Zbl 1163.60036 |
| [22] | Borodin, A.; Olshanski, G., The boundary of the Gelfand-Tsetlin graph: a new approach, Adv. Math., 230, 1738-1779 (2012) · Zbl 1245.05131 |
| [23] | Borodin, A.; Olshanski, G., The Young bouquet and its boundary, Mosc. Math. J., 13, 2, 193-232 (2013) · Zbl 1320.05137 |
| [24] | Borodin, A.; Olshanski, G., Representations of the Infinite Symmetric Group (2016), Cambridge: Cambridge University Press, Cambridge · Zbl 1364.20001 |
| [25] | Borodin, A.; Petrov, L., Integrable probability: from representation theory to Macdonald processes, Probab. Surv., 11, 1-58 (2014) · Zbl 1295.82023 |
| [26] | Borodin, A.; Petrov, L., Nearest neighbor Markov dynamics on Macdonald processes, Adv. Math., 300, 71-155 (2016) · Zbl 1356.60161 |
| [27] | Borodin, A., Wheeler, M.: Spin \(q\)-Whittaker polynomials (2017). arXiv preprint arXiv:1701.06292 [math.CO] · Zbl 1460.81033 |
| [28] | Brankov, J.; Priezzhev, V.; Schadschneider, A.; Schreckenberg, M., The Kasteleyn model and a cellular automaton approach to traffic flow, J. Phys. A Math. Gen., 29, 10, L229-L235 (1996) · Zbl 0901.60097 |
| [29] | Bufetov, A., Mucciconi, M., Petrov, L.: Yang-Baxter random fields and stochastic vertex models. Adv. Math. (2019). arXiv preprint arXiv:1905.06815 [math.PR] · Zbl 1469.05162 |
| [30] | Bufetov, A.; Petrov, L., Yang-Baxter field for spin Hall-Littlewood symmetric functions, Forum Math. Sigma, 7, e39 (2019) · Zbl 1480.60285 |
| [31] | Chhita, S.; Ferrari, P.; Spohn, H., Limit distributions for KPZ growth models with spatially homogeneous random initial conditions, Ann. Appl. Probab., 28, 3, 1573-1603 (2018) · Zbl 1397.82040 |
| [32] | Cohn, H.; Kenyon, R.; Propp, J., A variational principle for domino tilings, J. AMS, 14, 2, 297-346 (2001) · Zbl 1037.82016 |
| [33] | Corwin, I., The Kardar-Parisi-Zhang equation and universality class, Random Matrices Theory Appl., 1, 1130001 (2012) · Zbl 1247.82040 |
| [34] | Corwin, I.: The \(q\)-Hahn Boson process and \(q\)-Hahn TASEP. Int. Math. Res. Notices rnu094 (2014). arXiv:1401.3321 [math.PR] · Zbl 1335.82018 |
| [35] | Corwin, I.; Hammond, A., Brownian Gibbs property for Airy line ensembles, Invent. Math., 195, 2, 441-508 (2014) · Zbl 1459.82117 |
| [36] | Dauvergne, D., Ortmann, J., Virag, B.: The directed landscape (2018). arXiv preprint arXiv:1812.00309 [math.PR] |
| [37] | Di Francesco, P.; Guitter, E., A tangent method derivation of the arctic curve for q-weighted paths with arbitrary starting points, J. Phys. A, 52, 11, 115205 (2019) · Zbl 1507.82014 |
| [38] | Dynkin, E., Sufficient statistics and extreme points, Ann. Probab., 6, 705-730 (1978) · Zbl 0403.62009 |
| [39] | Ferrari, P.: The universal \(Airy_1\) and \(Airy_2\) processes in the Totally Asymmetric Simple Exclusion Process. In: Baik, J., Kriecherbauer, T., Li, L.-C., McLaughlin, K.T.-R., Tomei, C. (eds.) Integrable Systems and Random Matrices: In Honor of Percy Deift. Contemporary Math., pp. 321-332. AMS (2008). arXiv:math-ph/0701021 |
| [40] | Ferrari, P.; Occelli, A., Time-time covariance for last passage percolation with generic initial profile, Math. Phys. Anal. Geom., 22, 22, 1 (2019) · Zbl 1405.60144 |
| [41] | Ferrari, P.; Spohn, H., On time correlations for KPZ growth in one dimension, SIGMA, 12, 74 (2016) · Zbl 1344.60095 |
| [42] | Ferrari, PA, Limit theorems for tagged particles, Markov Process. Relat. Fields, 2, 1, 17-40 (1996) · Zbl 0879.60109 |
| [43] | Ferrari, PA, TASEP hydrodynamics using microscopic characteristics, Probab. Surv., 15, 1-27 (2018) · Zbl 1434.60288 |
| [44] | Ferrari, P.A., Martin, J.: Multiclass processes, dual points and M/M/1 queues (2005). arXiv preprint arXiv:math-ph/0509045 |
| [45] | Fulton, W.: Young Tableaux with Applications to Representation Theory and Geometry. Cambridge University Press. ISBN: 0521567246 (1997) · Zbl 0878.14034 |
| [46] | Gorin, V., The q-Gelfand-Tsetlin graph, Gibbs measures and q-Toeplitz matrices, Adv. Math., 229, 1, 201-266 (2012) · Zbl 1235.22028 |
| [47] | Gorin, V.; Olshanski, G., A quantization of the harmonic analysis on the infinite-dimensional unitary group, J. Funct. Anal., 270, 1, 375-418 (2016) · Zbl 1328.43007 |
| [48] | Guiol, H., Un résultat pour le processus d’exclusion à longue portée [A result for the long-range exclusion process], Ann. l’Institut Henri Poincare (B) Probab. Stat., 33, 4, 387-405 (1997) · Zbl 1002.60584 |
| [49] | Hammersley, J.M.: A few seedlings of research. In: Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, pp. 345-394 (1972) · Zbl 0236.00018 |
| [50] | Johansson, K., Shape fluctuations and random matrices, Commun. Math. Phys., 209, 2, 437-476 (2000) · Zbl 0969.15008 |
| [51] | Johansson, K.: The two-time distribution in geometric last-passage percolation (2018). arXiv preprint arXiv:1802.00729 [math.PR] · Zbl 1423.60158 |
| [52] | Johansson, K.: The long and short time asymptotics of the two-time distribution in local random growth (2019). arXiv preprint arXiv:1904.08195 [math.PR] |
| [53] | Johansson, K., Rahman, M.: Multi-time distribution in discrete polynuclear growth. Commun. Pure Appl. Math. (2020). arXiv:1906.01053 [math.PR] · Zbl 1530.60084 |
| [54] | Kenyon, R.; Okounkov, A., Limit shapes and the complex Burgers equation, Acta Math., 199, 2, 263-302 (2007) · Zbl 1156.14029 |
| [55] | Kerov, S.; Okounkov, A.; Olshanski, G., The boundary of Young graph with Jack edge multiplicities, Int. Math. Res. Notices, 4, 173-199 (1998) · Zbl 0960.05107 |
| [56] | Li, H., Petrov, L.: Computer simulation of the backwards TASEP evolution. https://lpetrov.cc/simulations/2019-06-26-back-tasep/ (2019) |
| [57] | Liggett, T., Interacting Particle Systems (2005), Berlin: Springer, Berlin · Zbl 1103.82016 |
| [58] | Liggett, TM, Ergodic theorems for the asymmetric simple exclusion process, Trans. Am. Math. Soc., 213, 237-261 (1975) · Zbl 0322.60086 |
| [59] | MacDonald, C.; Gibbs, J., Concerning the kinetics of polypeptide synthesis on polyribosomes, Biopolymers, 7, 5, 707-725 (1969) |
| [60] | MacDonald, C.; Gibbs, J.; Pipkin, A., Kinetics of biopolymerization on nucleic acid templates, Biopolymers, 6, 1, 1-25 (1968) |
| [61] | Macdonald, IG, Symmetric Functions and Hall Polynomials (1995), Oxford: Oxford University Press, Oxford · Zbl 0824.05059 |
| [62] | Matetski, K., Quastel, J., Remenik, D.: The KPZ fixed point (2017). arXiv preprint arXiv:1701.00018 [math.PR] · Zbl 1505.82041 |
| [63] | Mkrtchyan, S.; Petrov, L., GUE corners limit of q-distributed lozenge tilings, Electron. J. Probab., 22, 101, 24 (2017) · Zbl 1386.60035 · doi:10.1214/17-EJP112 |
| [64] | O’Connell, N., A path-transformation for random walks and the Robinson-Schensted correspondence, Trans. AMS, 355, 9, 3669-3697 (2003) · Zbl 1031.05132 |
| [65] | O’Connell, N., Conditioned random walks and the RSK correspondence, J. Phys. A, 36, 12, 3049-3066 (2003) · Zbl 1035.05097 |
| [66] | Okounkov, A., Infinite wedge and random partitions, Sel. Math., 7, 1, 57-81 (2001) · Zbl 0986.05102 |
| [67] | Okounkov, A.; Reshetikhin, N., Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram, J. AMS, 16, 3, 581-603 (2003) · Zbl 1009.05134 |
| [68] | Petrov, L., A two-parameter family of infinite-dimensional diffusions in the Kingman simplex, Funct. Anal. Appl., 43, 4, 279-296 (2009) · Zbl 1204.60076 |
| [69] | Petrov, L., The boundary of the Gelfand-Tsetlin graph: new proof of Borodin-Olshanski’s formula, and its q-analogue, Mosc. Math. J., 14, 1, 121-160 (2014) · Zbl 1297.05249 |
| [70] | Petrov, L., Zhang, E.: Computer simulations of dynamics on \(q\)-vol lozenge tilings inverting the parameter \(q\). https://lpetrov.cc/simulations/2019-04-30-qvol/ (2019) |
| [71] | Povolotsky, A., On integrability of zero-range chipping models with factorized steady state, J. Phys. A, 46, 465205 (2013) · Zbl 1290.82022 |
| [72] | Quastel, J.; Spohn, H., The one-dimensional KPZ equation and its universality class, J. Stat. Phys., 160, 4, 965-984 (2015) · Zbl 1327.82069 |
| [73] | Rákos, A.; Schütz, G., Current distribution and random matrix ensembles for an integrable asymmetric fragmentation process, J. Stat. Phys., 118, 3-4, 511-530 (2005) · Zbl 1126.82330 |
| [74] | Reshetikhin, N.: Lectures on the integrability of the 6-vertex model. In: Exact Methods in Low-Dimensional Statistical Physics and Quantum Computing, pp. 197-266. Oxford Univ. Press (2010). arXiv:1010.5031 [math-ph] · Zbl 1202.82022 |
| [75] | Romik, D., The Surprising Mathematics of Longest Increasing Subsequences (2015), Cambridge: Cambridge University Press, Cambridge · Zbl 1345.05003 |
| [76] | Rost, H., Nonequilibrium behaviour of a many particle process: density profile and local equilibria, Z. Wahrsch. Verw. Gebiete, 58, 1, 41-53 (1981) · Zbl 0451.60097 · doi:10.1007/BF00536194 |
| [77] | Sasamoto, T.; Wadati, M., Exact results for one-dimensional totally asymmetric diffusion models, J. Phys. A, 31, 6057-6071 (1998) · Zbl 1085.83501 |
| [78] | Seppäläinen, T., Existence of hydrodynamics for the totally asymmetric simple K-exclusion process, Ann. Probab., 27, 1, 361-415 (1999) · Zbl 0947.60088 |
| [79] | Sheffield, S.: Random surfaces. Astérisque 304 (2005). arXiv:math/0304049 [math.PR] · Zbl 1104.60002 |
| [80] | Spitzer, F., Interaction of Markov processes, Adv. Math., 5, 2, 246-290 (1970) · Zbl 0312.60060 |
| [81] | Spohn, H.: KPZ scaling theory and the semi-discrete directed polymer model (2012). arXiv:1201.0645 [cond-mat.stat-mech] |
| [82] | Takeyama, Y., A deformation of affine Hecke algebra and integrable stochastic particle system, J. Phys. A, 47, 46, 465203 (2014) · Zbl 1310.81083 |
| [83] | Toninelli, F., A \((2 + 1)\)-dimensional growth process with explicit stationary measures, Ann. Probab., 45, 5, 2899-2940 (2017) · Zbl 1383.82031 |
| [84] | Vershik, A.; Kerov, S., Asymptotic theory of the characters of the symmetric group, Funktsional. Anal. i Prilozhen., 15, 4, 15-27, 96 (1981) · Zbl 0507.20006 |
| [85] | Vershik, A.; Kerov, S., The characters of the infinite symmetric group and probability properties of the Robinson-Shensted-Knuth algorithm, SIAM J. Algebr. Discrete Math., 7, 1, 116-124 (1986) · Zbl 0584.05004 |
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