no. 5
Random permutations and unique fully supported ergodicity for the Euler adic transformation
Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 5, pp. 876-885.

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.

DOI : 10.1214/07-AIHP133
Classification : 37A05, 37A25, 37A50, 37B99, 60B05, 62F07
Mots clés : random permutations, eulerian numbers, adic transformation, invariant measures, ergodic transformations, Bratteli diagrams, rises and falls 
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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) no. 5, pp. 876-885. doi : 10.1214/07-AIHP133. http://www.numdam.org/articles/10.1214/07-AIHP133/

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

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

[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

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