Nous trouvons les noyaux de transition de quatre systèmes markoviens de particules en interaction sur une ligne, en prouvant que chacun de ces noyaux s'entrelace avec un noyau du type de Karlin-McGregor. Tous les noyaux résultants héritent de la structure de déterminant de la formule de Karlin-McGregor et ont une forme similaire à celle du noyau de Schütz pour le processus d'exclusion simple totalement asymétrique.
We find the transition kernels for four markovian interacting particle systems on the line, by proving that each of these kernels is intertwined with a Karlin-McGregor-type kernel. The resulting kernels all inherit the determinantal structure from the Karlin-McGregor formula, and have a similar form to Schütz's kernel for the totally asymmetric simple exclusion process.
Mots clés : interacting particle system, intertwining, Karlin-McGregor theorem, Markov transition kernel, Robinson-Schensted-Knuth correspondence, Schütz theorem, stochastic recursion, symmetric functions
@article{AIHPB_2008__44_6_1162_0, author = {Dieker, A. B. and Warren, J.}, title = {Determinantal transition kernels for some interacting particles on the line}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {1162--1172}, publisher = {Gauthier-Villars}, volume = {44}, number = {6}, year = {2008}, doi = {10.1214/07-AIHP176}, mrnumber = {2469339}, zbl = {1181.60144}, language = {en}, url = {http://www.numdam.org/articles/10.1214/07-AIHP176/} }
TY - JOUR AU - Dieker, A. B. AU - Warren, J. TI - Determinantal transition kernels for some interacting particles on the line JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2008 SP - 1162 EP - 1172 VL - 44 IS - 6 PB - Gauthier-Villars UR - http://www.numdam.org/articles/10.1214/07-AIHP176/ DO - 10.1214/07-AIHP176 LA - en ID - AIHPB_2008__44_6_1162_0 ER -
%0 Journal Article %A Dieker, A. B. %A Warren, J. %T Determinantal transition kernels for some interacting particles on the line %J Annales de l'I.H.P. Probabilités et statistiques %D 2008 %P 1162-1172 %V 44 %N 6 %I Gauthier-Villars %U http://www.numdam.org/articles/10.1214/07-AIHP176/ %R 10.1214/07-AIHP176 %G en %F AIHPB_2008__44_6_1162_0
Dieker, A. B.; Warren, J. Determinantal transition kernels for some interacting particles on the line. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 6, pp. 1162-1172. doi : 10.1214/07-AIHP176. http://www.numdam.org/articles/10.1214/07-AIHP176/
[1] Exact solution of a one-parameter family of asymmetric exclusion processes. Phys. Rev. E 57 (1998) 6370-6376. | MR
, and .[2] GUEs and queues. Probab. Theory Related Fields 119 (2001) 256-274. | MR | Zbl
.[3] Large time asymptotics of growth models on space-like paths I: PushASEP. Available at arXiv.org/abs/0707. 2813, 2007. | MR
and .[4] Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129 (2007) 1055-1080. | MR | Zbl
, , and .[5] Transition probabilities for series Jackson networks. Preprint, 2007.
and .[6] Queues, stores, and tableaux. J. Appl. Probab. 42 (2005) 1145-1167. | MR
, and .[7] Young Tableaux. Cambridge University Press, 1997. | MR | Zbl
.[8] Matrix correspondences of plane partitions. Pacific J. Math. 92 (1981) 295-315. | MR | Zbl
.[9] Shape fluctuations and random matrices. Comm. Math. Phys. 209 (2000) 437-476. | MR | Zbl
.[10] A multi-dimensional Markov chain and the Meixner ensemble. Available at arXiv.org/abs/0707.0098, 2007.
.[11] Orthogonal polynomial ensembles in probability theory. Probab. Surv. 2 (2005) 385-447. | MR
.[12] Conditioned random walks and the RSK correspondence. J. Phys. A 36 (2003) 3049-3066. | MR | Zbl
.[13] A path-transformation for random walks and the Robinson-Schensted correspondence. Trans. Amer. Math. Soc. 355 (2003) 3669-3697. | MR | Zbl
.[14] Determinant solution for the totally asymmetric exclusion process with parallel update. J. Stat. Mech. (2006) P07002.
and .[15] Current distribution and random matrix ensembles for an integrable asymmetric fragmentation process. J. Stat. Phys. 118 (2005) 511-530. | MR | Zbl
and .[16] Bethe ansatz and current distribution for the TASEP with particle-dependent hopping rates. Markov Process. Related Fields 12 (2006) 323-334. | MR | Zbl
and .[17] Exact solution of the master equation for the asymmetric exclusion process. J. Stat. Phys. 88 (1997) 427-445. | MR | Zbl
.[18] Exact limiting shape for a simplified model of first-passage percolation on the plane. Ann. Probab. 26 (1998) 1232-1250. | MR | Zbl
.[19] Enumerative Combinatorics, Vol. 1. Cambridge University Press, 1997. | MR | Zbl
.[20] Enumerative Combinatorics, Vol. 2. Cambridge University Press, 1999. | MR | Zbl
.[21] Integral formulas for the asymmetric simple exclusion process. Comm. Math. Phys. 279 (2008) 815-844. | MR | Zbl
and .[22] Dyson's Brownian motions, intertwining and interlacing. Electron. J. Probab. 12 (2007) 573-590. | MR | Zbl
.Cité par Sources :