Green functions for killed random walks in the Weyl chamber of Sp(4)

Raschel, Kilian

Annales de l'I.H.P. Probabilités et statistiques, Tome 47 (2011) no. 4, pp. 1001-1019.

Dans cet article, nous considérons une famille de marches aléatoires tuées au bord de la chambre de Weyl du dual de Sp(4), qui vérifie en outre la propriété suivante : pour tout n ≥ 3, il y a, dans cette famille, une marche ayant un groupe de réflexions d'ordre 2n. De plus, le cas n = 4 correspond à un processus bien connu apparaissant lors de l'étude des marches aléatoires quantiques sur le dual de Sp(4). Pour tous les processus de cette famille, nous trouvons l'asymptotique exacte des fonctions de Green selon toutes les trajectoires, ainsi que l'asymptotique des probabilités d'absorption sur le bord.

We consider a family of random walks killed at the boundary of the Weyl chamber of the dual of Sp(4), which in addition satisfies the following property: for any n ≥ 3, there is in this family a walk associated with a reflection group of order 2n. Moreover, the case n = 4 corresponds to a process which appears naturally by studying quantum random walks on the dual of Sp(4). For all the processes belonging to this family, we find the exact asymptotic of the Green functions along all infinite paths of states as well as that of the absorption probabilities along the boundaries.

[1] A. Ancona. Principe de Harnack à la frontière et théorème de Fatou pour un opérateur elliptique dans un domaine lipschitzien. Ann. Inst. Fourier (Grenoble) 28 (1978) 169-213. | Numdam | MR | Zbl

[2] R. Bañuelos and R. Smits. Brownian motion in cones. Probab. Theory Related Fields 108 (1997) 299-319. | MR | Zbl

[3] P. Biane. Quantum random walk on the dual of SU (n). Probab. Theory Related Fields 89 (1991) 117-129. | MR | Zbl

[4] P. Biane. Minuscule weights and random walks on lattices. In Quantum Probability and Related Topics 51-65. World Sci. Publ., River Edge, NJ, 1992. | MR | Zbl

[5] N. Bourbaki. Éléments de mathématique. Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées. Hermann, Paris, 1975. | MR | Zbl

[6] M. Bousquet-Mélou and M. Mishna. Walks with small steps in the quarter plane. In Algorithmic Probability and Combinatorics 1-40. Amer. Math. Soc., Providence, RI, 2010. | MR | Zbl

[7] B. Collins. Martin boundary theory of some quantum random walks. Ann. Inst. H. Poincaré Probab. Statist. 40 (2004) 367-384. | Numdam | MR | Zbl

[8] D. Denisov and V. Wachtel. Conditional limit theorems for ordered random walks. Electron. J. Probab. 15 (2010) 292-322. | MR | Zbl

[9] Y. Doumerc and N. O'Connell. Exit problems associated with finite reflection groups. Probab. Theory Related Fields 132 (2005) 501-538. | MR | Zbl

[10] E. Dynkin. The boundary theory of Markov processes (discrete case). Uspehi Mat. Nauk 24 (1969) 3-42. | MR | Zbl

[11] F. Dyson. A Brownian-motion model for the eigenvalues of a random matrix. J. Math. Phys. 3 (1962) 1191-1198. | MR | Zbl

[12] P. Eichelsbacher and W. König. Ordered random walks. Electron. J. Probab. 13 (2008) 1307-1336. | MR | Zbl

[13] G. Fayolle, R. Iasnogorodski and V. Malyshev. Random Walks in the Quarter-Plane. Springer, Berlin, 1999. | MR | Zbl

[14] I. Ignatiouk-Robert. Martin boundary of a killed random walk on ℤ+d. Preprint, UMR CNRS 8088, Universite de Cergy-Pontoise, 2009.

[15] I. Ignatiouk-Robert. Martin boundary of a reflected random walk on a half-space. Probab. Theory Related Fields 148 (2010) 197-245. | MR | Zbl

[16] I. Ignatiouk-Robert and C. Loree. Martin boundary of a killed random walk on a quadrant. Ann. Probab. 38 (2010) 1106-1142. | MR | Zbl

[17] G. Jones and D. Singerman. Complex Functions. Cambridge Univ. Press, Cambridge, 1987. | MR | Zbl

[18] H. Kesten. Hitting probabilities of random walks on ℤd. Stochastic Process. Appl. 25 (1987) 165-184. | MR | Zbl

[19] J. Komlós, P. Major and G. Tusnády. An approximation of partial sums of independent RV 's and the sample DF . I. Z. Wahrsch. Verw. Gebiete 32 (1975) 111-131. | MR | Zbl

[20] J. Komlós, P. Major and G. Tusnády. An approximation of partial sums of independent RV's, and the sample DF. II. Z. Wahrsch. Verw. Gebiete 34 (1976) 33-58. | MR | Zbl

[21] W. König and P. Schmid. Random walks conditioned to stay in Weyl chambers of type C and D. Electron. Comm. Probab. 15 (2010) 286-296. | MR

[22] M. Kozdron and G. Lawler. Estimates of random walk exit probabilities and application to loop-erased random walk. Electron. J. Probab. 10 (2005) 1442-1467. | MR | Zbl

[23] I. Kurkova and K. Raschel. Random walks in ℤ+2 with non-zero drift absorbed at the axes. Bull. Soc. Math. France. To appear.

[24] G. Lawler and V. Limic. The Beurling estimate for a class of random walks. Electron. J. Probab. 9 (2004) 846-861. | MR | Zbl

[25] G. Lawler and V. Limic. Random Walk: A Modern Introduction. Cambridge Univ. Press, Cambridge, 2010. | MR | Zbl

[26] M. Picardello and W. Woess. Martin boundaries of cartesian products of Markov chains. Nagoya Math. J. 128 (1992) 153-169. | MR | Zbl

[27] K. Raschel. Chemins confinés dans un quadrant. Thèse de doctorat de l'Université Pierre et Marie Curie, 2010.

[28] K. Raschel. Green functions and Martin compactification for killed random walks related to SU(3). Electron. Comm. Probab. 15 (2010) 176-190. | MR

[29] K. Uchiyama. The Green functions of two dimensional random walks killed on a line and their higher dimensional analogues. Electron. J. Probab. 15 (2010) 1161-1189. | MR

[30] K. Uchiyama. Random walks on the upper half plane. Preprint, Tokyo Institute of Technology, 2010.