Determinantal transition kernels for some interacting particles on the line
Annales de l'I.H.P. Probabilités et statistiques, Volume 44 (2008) no. 6, p. 1162-1172

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.

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.

DOI : https://doi.org/10.1214/07-AIHP176
Classification:  60J05,  60K35,  05E10,  05E05,  15A52
Keywords: 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, Jonathan},
     title = {Determinantal transition kernels for some interacting particles on the line},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {44},
     number = {6},
     year = {2008},
     pages = {1162-1172},
     doi = {10.1214/07-AIHP176},
     zbl = {1181.60144},
     mrnumber = {2469339},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 44 (2008) no. 6, pp. 1162-1172. doi : 10.1214/07-AIHP176. http://www.numdam.org/item/AIHPB_2008__44_6_1162_0/

[1] M. Alimohammadi, V. Karimipour and M. Khorrami. Exact solution of a one-parameter family of asymmetric exclusion processes. Phys. Rev. E 57 (1998) 6370-6376. | MR 1628226

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

[3] A. Borodin and P. L. Ferrari. Large time asymptotics of growth models on space-like paths I: PushASEP. Available at arXiv.org/abs/0707. 2813, 2007. | MR 2438811

[4] A. Borodin, P. L. Ferrari, M. Prähofer and T. Sasamoto. Fluctuation properties of the TASEP with periodic initial configuration. J. Stat. Phys. 129 (2007) 1055-1080. | MR 2363389 | Zbl 1136.82028

[5] A. B. Dieker and J. Warren. Transition probabilities for series Jackson networks. Preprint, 2007.

[6] M. Draief, J. Mairesse and N. O'Connell. Queues, stores, and tableaux. J. Appl. Probab. 42 (2005) 1145-1167. | MR 2203829 | Zbl pre05036960

[7] W. Fulton. Young Tableaux. Cambridge University Press, 1997. | MR 1464693 | Zbl 0878.14034

[8] E. R. Gansner. Matrix correspondences of plane partitions. Pacific J. Math. 92 (1981) 295-315. | MR 618067 | Zbl 0432.05010

[9] K. Johansson. Shape fluctuations and random matrices. Comm. Math. Phys. 209 (2000) 437-476. | MR 1737991 | Zbl 0969.15008

[10] K. Johansson. A multi-dimensional Markov chain and the Meixner ensemble. Available at arXiv.org/abs/0707.0098, 2007.

[11] W. König. Orthogonal polynomial ensembles in probability theory. Probab. Surv. 2 (2005) 385-447. | MR 2203677

[12] N. O'Connell. Conditioned random walks and the RSK correspondence. J. Phys. A 36 (2003) 3049-3066. | MR 1986407 | Zbl 1035.05097

[13] 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

[14] A. M. Povolotsky and V. B. Priezzhev. Determinant solution for the totally asymmetric exclusion process with parallel update. J. Stat. Mech. (2006) P07002.

[15] A. Rákos and G. Schütz. Current distribution and random matrix ensembles for an integrable asymmetric fragmentation process. J. Stat. Phys. 118 (2005) 511-530. | MR 2123646 | Zbl 1126.82330

[16] 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. | MR 2249635 | Zbl 1136.82350

[17] G. M. Schütz. Exact solution of the master equation for the asymmetric exclusion process. J. Stat. Phys. 88 (1997) 427-445. | MR 1468391 | Zbl 0945.82508

[18] T. Seppäläinen. Exact limiting shape for a simplified model of first-passage percolation on the plane. Ann. Probab. 26 (1998) 1232-1250. | MR 1640344 | Zbl 0935.60093

[19] R. P. Stanley. Enumerative Combinatorics, Vol. 1. Cambridge University Press, 1997. | MR 1442260 | Zbl 0889.05001

[20] R. P. Stanley. Enumerative Combinatorics, Vol. 2. Cambridge University Press, 1999. | MR 1676282 | Zbl 0928.05001

[21] C. A. Tracy and H. Widom. Integral formulas for the asymmetric simple exclusion process. Comm. Math. Phys. 279 (2008) 815-844. | MR 2386729 | Zbl 1148.60080

[22] J. Warren. Dyson's Brownian motions, intertwining and interlacing. Electron. J. Probab. 12 (2007) 573-590. | MR 2299928 | Zbl 1127.60078