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2017 Transport-entropy inequalities on locally acting groups of permutations
Paul-Marie Samson
Electron. J. Probab. 22: 1-33 (2017). DOI: 10.1214/17-EJP54

Abstract

Following Talagrand’s concentration results for permutations picked uniformly at random from a symmetric group [27], Luczak and McDiarmid have generalized it to more general groups of permutations which act suitably ‘locally’. Here we extend their results by setting transport-entropy inequalities on these permutations groups. Talagrand and Luczak-Mc-Diarmid concentration properties are consequences of these inequalities. The results are also generalised to a larger class of measures including Ewens distributions of arbitrary parameter $\theta $ on the symmetric group. By projection, we derive transport-entropy inequalities for the uniform law on the slice of the discrete hypercube and more generally for the multinomial law. These results are new examples, in discrete setting, of weak transport-entropy inequalities introduced in [7], that contribute to a better understanding of the concentration properties of measures on permutations groups. One typical application is deviation bounds for the so-called configuration functions, such as the number of cycles of given lenght in the cycle decomposition of a random permutation.

Citation

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Paul-Marie Samson. "Transport-entropy inequalities on locally acting groups of permutations." Electron. J. Probab. 22 1 - 33, 2017. https://doi.org/10.1214/17-EJP54

Information

Received: 1 December 2016; Accepted: 31 March 2017; Published: 2017
First available in Project Euclid: 9 August 2017

zbMATH: 1380.60030
MathSciNet: MR3690287
Digital Object Identifier: 10.1214/17-EJP54

Subjects:
Primary: 26D10, 32F32, ‎39B62, 60E15

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