Limits of multilevel TASEP and similar processes
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, p. 18-27
Nous étudions le comportement asymptotique d’une classe de dynamiques aléatoires sur des configurations entrelacées de particules (dites aussi motifs de Gelfand–Tsetlin). Des exemples de telles dynamiques incluent, en particulier, une extension à plusieurs niveaux du TASEP et des dynamiques de particules reliées à l’algorithme de mélange pour les pavages par dominos du diamant aztèque. Nous montrons que le processus des mouvements browniens réfléchis entrelacés introduit par Warren dans (Electron. J. Probab. 12 (2007) 573–590) est une limite d’échelle universelle pour ces dynamiques.
We study the asymptotic behavior of a class of stochastic dynamics on interlacing particle configurations (also known as Gelfand–Tsetlin patterns). Examples of such dynamics include, in particular, a multi-layer extension of TASEP and particle dynamics related to the shuffling algorithm for domino tilings of the Aztec diamond. We prove that the process of reflected interlacing Brownian motions introduced by Warren in (Electron. J. Probab. 12 (2007) 573–590) serves as a universal scaling limit for such dynamics.
DOI : https://doi.org/10.1214/13-AIHP555
Classification:  60J27,  60K35,  60F17
@article{AIHPB_2015__51_1_18_0,
     author = {Gorin, Vadim and Shkolnikov, Mykhaylo},
     title = {Limits of multilevel TASEP and similar processes},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {1},
     year = {2015},
     pages = {18-27},
     doi = {10.1214/13-AIHP555},
     zbl = {06412896},
     mrnumber = {3300962},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2015__51_1_18_0}
}
Gorin, Vadim; Shkolnikov, Mykhaylo. Limits of multilevel TASEP and similar processes. Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, pp. 18-27. doi : 10.1214/13-AIHP555. http://www.numdam.org/item/AIHPB_2015__51_1_18_0/

[1] Y. Baryshnikov. GUEs and queues. Probab. Theory Related Fields 119 (2) (2001) 256–274. | MR 1818248 | Zbl 0980.60042

[2] D. Betea. Elliptically distributed lozenge tilings of a hexagon. Available at arXiv:1110.4176.

[3] A. Borodin and P. Ferrari. Large time asymptotics of growth models on space-like paths I: PushASEP. Electron. J. Probab. 13 (2008) 1380–1418. Available at arXiv:0707.2813. | MR 2438811 | Zbl 1187.82084

[4] A. Borodin and P. Ferrari. Anisotropic growth of random surfaces in 2 + 1 dimensions. Comm. Math. Phys. 325 (2014) 603–684. Available at arXiv:0804.3035. | MR 3148098 | Zbl 1303.82015

[5] A. Borodin and V. Gorin. Shuffling algorithm for boxed plane partitions. Adv. Math. 220 (6) (2009) 1739–1770. Available at arXiv:0804.3071. | MR 2493180 | Zbl 1172.60020

[6] A. Borodin and V. Gorin. Markov processes of infinitely many nonintersecting random walks. Probab. Theory Related Fields 155 (2013) 935–997. Available at arXiv:1106.1299. | MR 3034797 | Zbl 1278.60113

[7] A. Borodin, V. Gorin and E. Rains. q-distributions on boxed plane partitions. Selecta Math. (N.S.) 16 (4) (2010) 731–789. Available at arXiv:0905.0679. | MR 2734330 | Zbl 1205.82122

[8] A. Borodin and J. Kuan. Asymptotics of Plancherel measures for the infinite-dimensional unitary group. Adv. Math. 219 (3) (2008) 894–931. Available at arXiv:0712.1848. | MR 2442056 | Zbl 1153.60058

[9] K. Burdzy, W. Kang and K. Ramanan. The Skorokhod problem in a time-dependent interval. Stoch. Process. Appl. 119 (2) (2009) 428–452. | MR 2493998 | Zbl 1186.60035

[10] P. Dupuis and K. Ramanan. Convex duality and the Skorokhod problem II. Probab. Theory Related Fields 115 (1999) 197–236. | MR 1720348 | Zbl 0944.60062

[11] N. Elkies, G. Kuperberg, M. Larsen and J. Propp. Alternating-sign matrices and domino tilings. II. J. Algebraic Combin. 1 (3) (1992) 219–234. | MR 1194076 | Zbl 0788.05017

[12] S. N. Ethier and T. G. Kurtz. Markov Processes: Characterization and Convergence. Wiley, New York, 1986. | MR 838085 | Zbl 1089.60005

[13] J. M. Harrison and M. I. Reiman. Reflected Brownian motion on an orthant. Ann. Probab. 9 (2) (1981) 302–308. | MR 606992 | Zbl 0462.60073

[14] K. Johansson. Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. of Math. (2) 153 (1) (2001) 259–296. | MR 1826414 | Zbl 0984.15020

[15] K. Johansson. The arctic circle boundary and the Airy process. Ann. Probab. 33 (1) (2005) 1–30. | MR 2118857 | Zbl 1096.60039

[16] K. Johansson and E. Nordenstam. Eigenvalues of GUE minors. Electron. J. Probab. 11 (2006) 1342–1371 (electronic). | MR 2268547 | Zbl 1127.60047

[17] E. Nordenstam. On the shuffling algorithm for domino tilings. Electon. J. Probab. 15 (2010) 75–95. Available at arXiv:0802.2592. | MR 2578383 | Zbl 1193.60015

[18] N. O’Connell. A path-transformation for random walks and the Robinson–Schensted correspondence. Trans. Amer. Math. Soc. 355 (2003) 3669–3697. | MR 1990168 | Zbl 1031.05132

[19] N. O’Connell. Conditioned random walks and the RSK correspondence. J. Phys. A: Math. Gen. 36 (2003) 3049–3066. | MR 1986407 | Zbl 1035.05097

[20] A. Yu. Okounkov and N. Yu. Reshetikhin. The birth of a random matrix. Mosc. Math. J. 6 (3) (2006) 553–566. | MR 2274865 | Zbl 1130.15014

[21] A. Rákos and G. Schütz. Bethe Ansatz and current distribution for the TASEP with particle-dependent hopping rates. Markov Process. Related Fields 12 (2006) 323–334. Available at arXiv:cond-mat/0506525. | MR 2249635 | Zbl 1136.82350

[22] J. Warren. Dyson’s Brownian motions, intertwining and interlacing. Electon. J. Probab. 12 (2007) 573–590. Available at arXiv:math/0509720. | MR 2299928 | Zbl 1127.60078

[23] J. Warren and P. Windridge. Some examples of dynamics for Gelfand Tsetlin patterns. Electon. J. Probab. 14 (2009) 1745–1769. Available at arXiv:0812.0022. | MR 2535012 | Zbl 1196.60135