Random permutations and unique fully supported ergodicity for the Euler adic transformation
Frick, Sarah Bailey ; Petersen, Karl
Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008), p. 876-885 / Harvested from Numdam
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Pour la transformation adique sur l'espace des chemins infinis dans le graphe associé aux nombres Euleriens, il n'existe qu'une seule mesure de probabilité ergodique invariante avec support total. Ce résultat peut justifier en partie une hypothèse fréquente sur l'équidistribution des permutations aléatoires.

There is only one fully supported ergodic invariant probability measure for the adic transformation on the space of infinite paths in the graph that underlies the eulerian numbers. This result may partially justify a frequent assumption about the equidistribution of random permutations.

Publié le : 2008-01-01
DOI : https://doi.org/10.1214/07-AIHP133
Classification:  37A05,  37A25,  37A50,  37B99,  60B05,  62F07 
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     title = {Random permutations and unique fully supported ergodicity for the Euler adic transformation},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     volume = {44},
     year = {2008},
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     doi = {10.1214/07-AIHP133},
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     language = {en},
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Frick, Sarah Bailey; Petersen, Karl. Random permutations and unique fully supported ergodicity for the Euler adic transformation. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) pp. 876-885. doi : 10.1214/07-AIHP133. http://gdmltest.u-ga.fr/item/AIHPB_2008__44_5_876_0/

[1] S. Bailey, M. Keane, K. Petersen and I. Salama. Ergodicity of the adic transformation on the Euler graph. Math. Proc. Cambridge Philos. Soc. 141 (2006) 231-238. | MR 2265871 | Zbl 1112.28012

[2] L. Carlitz, D. C. Kurtz, R. Scoville and O. P. Stackelberg. Asymptotic properties of Eulerian numbers. Z. Wahrsch. Verw. Gebiete 23 (1972) 47-54. | MR 309856 | Zbl 0226.60049

[3] L. Comtet. Advanced Combinatorics. D. Reidel Publishing Co., Dordrecht, enlarged edition, 1974. The Art of Finite and Infinite Expansions. | MR 460128 | Zbl 0283.05001

[4] C.-G. Esseen. On the application of the theory of probability to two combinatorial problems involving permutations. In Proceedings of the Seventh Conference on Probability Theory (Braşov, 1982). VNU Sci. Press, Utrecht, 1985. | MR 867425 | Zbl 0619.60013

[5] S. B. Frick. Limited scope adic transformations. In preparation. | Zbl 1175.37004 | Zbl pre05617771

[6] S. B. Frick. Dynamical properties of some non-stationary, non-simple Bratteli-Vershik systems. Ph.D. dissertation, Univ. North Carolina, Chapel Hill (2006).

[7] S. B. Frick and K. Petersen. Connections between adic transformations and random walks. In progress.

[8] J. C. Fu and W. Y. W. Lou. Joint distribution of rises and falls. Ann. Inst. Statist. Math. 52 (2000) 415-425. | MR 1794242 | Zbl 0980.62010

[9] J. C. Fu, W. Y. W. Lou and Y.-J. Wang. On the exact distributions of Eulerian and Simon Newcomb numbers associated with random permutations. Statist. Probab. Lett. 42 (1999) 115-125. | MR 1680086 | Zbl 1057.62503

[10] T. Giordano, I. F. Putnam and C. F. Skau. Topological orbit equivalence and C*-crossed products. J. Reine Angew. Math. 469 (1995) 51-111. | MR 1363826 | Zbl 0834.46053

[11] R. H. Herman, I. F. Putnam and C. F. Skau. Ordered Bratteli diagrams, dimension groups and topological dynamics. Internat. J. Math. 3 (1992) 827-864. | MR 1194074 | Zbl 0786.46053

[12] X. Méla. Dynamical properties of the Pascal adic and related systems. Ph.D. dissertation, Univ. North Carolina, Chapel Hill (2002).

[13] X. Méla and K. Petersen. Dynamical properties of the Pascal adic transformation. Ergodic Theory Dynam. Systems 25 (2005) 227-256. | MR 2122921 | Zbl 1069.37007

[14] D. S. Ornstein, D. J. Rudolph and B. Weiss. Equivalence of measure preserving transformations. Mem. Amer. Math. Soc. 37 (1982). | MR 653094 | Zbl 0504.28019

[15] G. Oshanin and R. Voituriez. Random walk generated by random permutations of {1, 2, 3, …, n+1}. J. Phys. A: Math. Gen. 37 (2004) 6221-6241. | MR 2073602 | Zbl 1056.60045

[16] K. Petersen and K. Schmidt. Symmetric Gibbs measures. Trans. Amer. Math. Soc. 349 (1997) 2775-2811. | MR 1422906 | Zbl 0873.28008

[17] A. M. Vershik. Description of invariant measures for the actions of some infinite-dimensional groups. Dokl. Akad. Nauk SSSR 218 (1974) 749-752. | MR 372161 | Zbl 0324.28014

[18] A. M. Vershik and S. V. Kerov. Asymptotic theory of characters of the symmetric group. Funkts. Anal. Prilozhen. 15 (1981) 15-27. | MR 639197 | Zbl 0507.20006

[19] A. M. Vershik and S. V. Kerov. Locally semisimple algebras, combinatorial theory and the K0-functor. J. Soviet Math. 38 (1987) 1701-1733. | Zbl 0623.46036