Second class particles and limit shapes of evacuation and sliding paths for random tableaux. (English) Zbl 1511.60021
Summary: We investigate two closely related setups. In the first one we consider a TASEP-style system of particles with specified initial and final configurations. The probability of each history of the system is assumed to be equal. We show that the rescaled trajectory of the second class particle converges (as the size of the system tends to infinity) to a random arc of an ellipse.
In the second setup we consider a uniformly random Young tableau of square shape and look for typical (in the sense of probability) sliding paths and evacuation paths in the asymptotic setting as the size of the square tends to infinity. We show that the probability distribution of such paths converges to a random meridian connecting the opposite corners of the square. We also discuss analogous results for non-square Young tableaux.
In the second setup we consider a uniformly random Young tableau of square shape and look for typical (in the sense of probability) sliding paths and evacuation paths in the asymptotic setting as the size of the square tends to infinity. We show that the probability distribution of such paths converges to a random meridian connecting the opposite corners of the square. We also discuss analogous results for non-square Young tableaux.
MSC:
| 60C05 | Combinatorial probability |
| 05E10 | Combinatorial aspects of representation theory |
| 60K35 | Interacting random processes; statistical mechanics type models; percolation theory |
| 82C22 | Interacting particle systems in time-dependent statistical mechanics |
Keywords:
second class particles; interacting particle systems; TASEP; random Young tableaux; limit shape; jeu de taquin; promotion; Schützenberger’s evacuation; square Young tableauxReferences:
| [1] | Omer Angel, Alexander E. Holroyd, Dan Romik, and B{\'a}lint Vir{\'a}g. Random sorting networks. Adv. Math. 215(2):839-868, 2007, DOI 10.1016/j.aim.2007.05.019, zbl 1132.60008, MR2355610, arxiv math/0609538 · Zbl 1132.60008 · doi:10.1016/j.aim.2007.05.019 |
| [2] | Albert Benassi and Jean-Pierre Fouque. Hydrodynamical limit for the asymmetric simple exclusion process. Ann. Probab. 15(2):546-560, 1987, DOI 10.1214/aop/1176992158, zbl 0623.60120, MR0885130 · Zbl 0623.60120 · doi:10.1214/aop/1176992158 |
| [3] | Philippe Biane. Representations of symmetric groups and free probability. Adv. Math. 138(1):126-181,1998, DOI 10.1006/aima.1998.1745, zbl 0927.20008, MR1644993 · Zbl 0927.20008 · doi:10.1006/aima.1998.1745 |
| [4] | Philippe Biane. Approximate factorization and concentration for characters of symmetric groups. Internat. Math. Res. Notices (4):179-192, 2001, DOI 10.1155/S1073792801000113, zbl 1106.20304, MR1813797, arxiv math/0006111 · Zbl 1106.20304 · doi:10.1155/S1073792801000113 |
| [5] | Patrick Billingsley. Convergence of probability measures, second edition. Wiley Series in Probability and Statistics. A Wiley-Interscience Publication. Wiley, New York,1999, zbl 0944.60003, MR1700749 · Zbl 0944.60003 |
| [6] | Johannes M. Burgers. A mathematical model illustrating the theory of turbulence. In Advances in Applied Mechanics, 171-199. edited by Richard von Mises and Theodore von K\'{a}rm\'{a}n. Academic Press, Inc., New York, 1948, MR0027195 |
| [7] | Tullio Ceccherini-Silberstein, Fabio Scarabotti, and Filippo Tolli. Representation theory of the symmetric groups. The Okounkov-Vershik approach, character formulas, and partition algebras. Cambridge Studies in Advanced Mathematics, 121. Cambridge University Press, Cambridge, 2010, zbl 1230.20002, MR2643487 |
| [8] | Rick Durrett. Probability: theory and examples, fourth edition. Cambridge Series in Statistical and Probabilistic Mathematics, 31. Cambridge University Press, Cambridge, 2010, DOI 10.1017/CBO9780511779398, zbl 1202.60001, MR2722836 · Zbl 1202.60001 · doi:10.1017/CBO9780511779398 |
| [9] | Duncan Dauvergne and B\'{a}lint Vir\'{a}g. Circular support in random sorting networks. Trans. Amer. Math. Soc. 373(3):1529-1553, 2020, DOI 10.1090/tran/7819, zbl 1455.60020, MR4068272, arxiv 1802.08933 · Zbl 1455.60020 · doi:10.1090/tran/7819 |
| [10] | Pablo A. Ferrari. Shock fluctuations in asymmetric simple exclusion. Probab. Theory Related Fields 91(1):81-101, 1992, DOI 10.1007/BF01194491, zbl 0744.60117, MR1142763 · Zbl 0744.60117 · doi:10.1007/BF01194491 |
| [11] | Pablo A. Ferrari and Luiz Renato G. Fontes. Current fluctuations for the asymmetric simple exclusion process. Ann. Probab. 22(2):820-832, 1994, DOI 10.1214/aop/1176988731, zbl 0806.60099, MR1288133 · Zbl 0806.60099 · doi:10.1214/aop/1176988731 |
| [12] | Pablo A. Ferrari and Claude Kipnis. Second class particles in the rarefaction fan. Ann. Inst. H. Poincar\'e Probab. Statist. 31(1):143-154, 1995, zbl 0813.60095, MR1340034 · Zbl 0813.60095 |
| [13] | Valentin F\'{e}ray and Piotr {\'S}niady. Asymptotics of characters of symmetric groups related to Stanley character formula. Ann. of Math. (2) 173(2):887-906, 2011, DOI 10.4007/annals.2011.173.2.6, zbl 1229.05276, MR2776364, arxiv math/0701051 · Zbl 1229.05276 · doi:10.4007/annals.2011.173.2.6 |
| [14] | William Fulton. Young tableaux. With applications to representation theory and geometry. London Mathematical Society Student Texts, 35. Cambridge University Press, Cambridge, 1997, zbl 0878.14034, MR1464693 · Zbl 0878.14034 |
| [15] | Curtis Greene. An extension of Schensted’s theorem. Advances in Math. 14:254-265, 1974, DOI 10.1016/0001-8708(74)90031-0, zbl 0303.05006, MR0354395 · Zbl 0303.05006 · doi:10.1016/0001-8708(74)90031-0 |
| [16] | A.-A. A. Jucys. Symmetric polynomials and the center of the symmetric group ring. Rep. Mathematical Phys. 5(1):107-112, 1974, DOI 10.1016/0034-4877(74)90019-6, zbl 0288.20014, MR0419576 · Zbl 0288.20014 · doi:10.1016/0034-4877(74)90019-6 |
| [17] | Sergei V. Kerov. The asymptotics of interlacing sequences and the growth of continual Young diagrams. Zap. Nauchn. Semin. POMI 205:21-29, 1993, zbl 0804.33018, MR1255301 · Zbl 0804.33018 |
| [18] | Sergei V. Kerov. Transition probabilities of continual Young diagrams and the Markov moment problem. Funktsional. Anal. i Prilozhen. 27(2):32-49, 1993, DOI 10.1007/BF01085981, zbl 0808.05098, MR1251166 · Zbl 0808.05098 · doi:10.1007/BF01085981 |
| [19] | Sergei V. Kerov. Interlacing measures. In Kirillov’s seminar on representation theory, Amer. Math. Soc. Transl. Ser. 2, 181, 35-83. Amer. Math. Soc., Providence, RI, 1998, zbl 0890.05074, MR1618739 |
| [20] | Richard Kenyon and Istv{\'a}n Prause. Gradient variational problems in \(\mathbb{R}^2\). Duke Math. J. 171(14):3003-3022, 2022, DOI 10.1215/00127094-2022-0036, zbl 07600557, MR4491711, arxiv 2006.01219 · Zbl 1502.49003 · doi:10.1215/00127094-2022-0036 |
| [21] | Thomas M. Liggett. Stochastic interacting systems: contact, voter and exclusion processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 324. Springer, Berlin, 1999, DOI 10.1007/978-3-662-03990-8, zbl 0949.60006, MR1717346 · Zbl 0949.60006 · doi:10.1007/978-3-662-03990-8 |
| [22] | Thomas Mountford and Herv{\'e} Guiol. The motion of a second class particle for the TASEP starting from a decreasing shock profile. Ann. Appl. Probab. 15(2):1227-1259, 2005, DOI 10.1214/105051605000000151, zbl 1069.60091, MR2134103, arxiv math/0505216 · Zbl 1069.60091 · doi:10.1214/105051605000000151 |
| [23] | James A. Mingo and Roland Speicher. Free probability and random matrices. Fields Institute Monographs, 35. Springer, New York; Fields Institute for Research in Mathematical Sciences, Toronto, ON, 2017, DOI 10.1007/978-1-4939-6942-5, zbl 1387.60005, MR3585560, arxiv 1404.3393 · Zbl 1387.60005 · doi:10.1007/978-1-4939-6942-5 |
| [24] | {\L}ukasz Ma\'{s}lanka and Piotr {\'S}niady. Limit shapes of evacuation and jeu de taquin paths in random square tableaux. S\'{e}m. Lothar. Combin. 84B:Art. 8, 12, 2020, zbl 1447.05225, MR4138636, arxiv 1911.08143 · Zbl 1447.05225 |
| [25] | Boris Pittel and Dan Romik. Limit shapes for random square Young tableaux. Adv. in Appl. Math. 38(2):164-209, 2007, DOI 10.1016/j.aam.2005.12.005, zbl 1122.60009, MR2290809, arxiv math/0405190 · Zbl 1122.60009 · doi:10.1016/j.aam.2005.12.005 |
| [26] | Steven Pon and Qiang Wang. Promotion and evacuation on standard {Y}oung tableaux of rectangle and staircase shape. Electron. J. Combin. 18(1):Paper 18, 18 p., 2011, zbl 1219.05198, MR2770123, arxiv 1003.2728 · Zbl 1219.05198 |
| [27] | Fraydoun Rezakhanlou. Microscopic structure of shocks in one conservation laws. Ann. Inst. H. Poincar\'{e} Anal. Non Lin\'{e}aire 12(2):119-153, 1995, DOI 10.1016/S0294-1449(16)30161-5, zbl 0836.76046, MR1326665 · Zbl 0836.76046 · doi:10.1016/S0294-1449(16)30161-5 |
| [28] | Dan Romik. Explicit formulas for hook walks on continual Young diagrams. Adv. in Appl. Math. 32(4):625-654, 2004, DOI 10.1016/S0196-8858(03)00096-4, zbl 1051.05080, MR2053837, arxiv math/0303376 · Zbl 1051.05080 · doi:10.1016/S0196-8858(03)00096-4 |
| [29] | Dan Romik. Permutations with short monotone subsequences. Adv. in Appl. Math. 37(4):501-510, 2006, DOI 10.1016/j.aam.2005.08.008, zbl 1109.05015, MR2266895 · Zbl 1109.05015 · doi:10.1016/j.aam.2005.08.008 |
| [30] | Dan Romik. Arctic circles, domino tilings and square Young tableaux. Ann. Probab. 40(2):611-647, 2012, DOI 10.1214/10-AOP628, zbl 1258.60014, MR2952086, arxiv 0910.1636 · Zbl 1258.60014 · doi:10.1214/10-AOP628 |
| [31] | Hermann Rost. Nonequilibrium behaviour of a many particle process: density profile and local equilibria. Z. Wahrsch. Verw. Gebiete 58(1):41-53, 1981, DOI 10.1007/BF00536194, zbl 0451.60097, MR0635270 · Zbl 0451.60097 · doi:10.1007/BF00536194 |
| [32] | Dan Romik and Piotr {\'S}niady. Jeu de taquin dynamics on infinite Young tableaux and second class particles. Ann. Probab. 43(2):682-737, 2015, DOI 10.1214/13-AOP873, zbl 1360.60028, MR3306003, arxiv 1111.0575 · Zbl 1360.60028 · doi:10.1214/13-AOP873 |
| [33] | Bruce E. Sagan. The symmetric group: Representations, combinatorial algorithms, and symmetric functions, second edition. Graduate Texts in Mathematics, 203. Springer, New York, 2001, DOI 10.1007/978-1-4757-6804-6, zbl 0964.05070, MR1824028 · Zbl 0964.05070 · doi:10.1007/978-1-4757-6804-6 |
| [34] | Craige E. Schensted. Longest increasing and decreasing subsequences. Canadian J. Math. 13:179-191, 1961, DOI 10.4153/CJM-1961-015-3, zbl 0097.25202, MR0121305 · Zbl 0097.25202 · doi:10.4153/CJM-1961-015-3 |
| [35] | Marcel Paul Sch{\"u}tzenberger. Quelques remarques sur une construction de Schensted. Math. Scand. 12:117-128, 1963, DOI 10.7146/math.scand.a-10676, zbl 0216.30202, MR0190017 · Zbl 0216.30202 · doi:10.7146/math.scand.a-10676 |
| [36] | Timo Sepp\"{a}l\"{a}inen. Second class particles as microscopic characteristics in totally asymmetric nearest-neighbor \(K\)-exclusion processes. Trans. Amer. Math. Soc. 353(12):4801-4829, 2001, DOI 10.1090/S0002-9947-01-02872-0, zbl 0983.60093, MR1852083, arxiv math/0009040 · Zbl 0983.60093 · doi:10.1090/S0002-9947-01-02872-0 |
| [37] | Piotr {\'S}niady. Gaussian fluctuations of characters of symmetric groups and of Young diagrams. Probab. Theory Related Fields 136(2):263-297, 2006, DOI 10.1007/s00440-005-0483-y, zbl 1104.46035, MR2240789, arxiv math/0501112 · Zbl 1104.46035 · doi:10.1007/s00440-005-0483-y |
| [38] | Piotr {\'S}niady. Robinson-Schensted-Knuth algorithm, jeu de taquin, and Kerov-Vershik measures on infinite tableaux. SIAM J. Discrete Math. 28(2):598-630, 2014, DOI 10.1137/130930169, zbl 1300.60025, MR3190750, arxiv 1307.5645 · Zbl 1300.60025 · doi:10.1137/130930169 |
| [39] | Frank Spitzer. Interaction of Markov processes. Advances in Math. 5:246-290, 1970, DOI 10.1016/0001-8708(70)90034-4, zbl 0312.60060, MR0268959 · Zbl 0312.60060 · doi:10.1016/0001-8708(70)90034-4 |
| [40] | Richard P. Stanley. Enumerative combinatorics. Vol. 2. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin. Cambridge Studies in Advanced Mathematics, 62. Cambridge University Press, Cambridge, 1999, zbl 0928.05001, MR1676282 · Zbl 0928.05001 |
| [41] | Wangru Sun. Dimer model, bead model and standard young tableaux: finite cases and limit shapes. Preprint, 2018, arxiv 1804.03414 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
