Limits of multilevel TASEP and similar processes
Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 1, pp. 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 : 10.1214/13-AIHP555
Classification : 60J27, 60K35, 60F17
Mots-clés : interacting particle system, exclusion process, reflected brownian motion 
@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},
     pages = {18--27},
     publisher = {Gauthier-Villars},
     volume = {51},
     number = {1},
     year = {2015},
     doi = {10.1214/13-AIHP555},
     mrnumber = {3300962},
     zbl = {06412896},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1214/13-AIHP555/}
}
TY  - JOUR
AU  - Gorin, Vadim
AU  - Shkolnikov, Mykhaylo
TI  - Limits of multilevel TASEP and similar processes
JO  - Annales de l'I.H.P. Probabilités et statistiques
PY  - 2015
SP  - 18
EP  - 27
VL  - 51
IS  - 1
PB  - Gauthier-Villars
UR  - http://archive.numdam.org/articles/10.1214/13-AIHP555/
DO  - 10.1214/13-AIHP555
LA  - en
ID  - AIHPB_2015__51_1_18_0
ER  - 
%0 Journal Article
%A Gorin, Vadim
%A Shkolnikov, Mykhaylo
%T Limits of multilevel TASEP and similar processes
%J Annales de l'I.H.P. Probabilités et statistiques
%D 2015
%P 18-27
%V 51
%N 1
%I Gauthier-Villars
%U http://archive.numdam.org/articles/10.1214/13-AIHP555/
%R 10.1214/13-AIHP555
%G en
%F 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://archive.numdam.org/articles/10.1214/13-AIHP555/

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

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

[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. | DOI | EuDML | MR | Zbl

[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. | DOI | MR | Zbl

[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 | Zbl

[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. | DOI | MR | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

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

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

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

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

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

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

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

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

[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 | Zbl

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

[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 | Zbl

Cité par Sources :