Electronic Journal of Probability

Transport-entropy inequalities on locally acting groups of permutations

Paul-Marie Samson

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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.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 62, 33 pp.

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

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Primary: 60E15: Inequalities; stochastic orderings 32F32: Analytical consequences of geometric convexity (vanishing theorems, etc.) 39B62: Functional inequalities, including subadditivity, convexity, etc. [See also 26A51, 26B25, 26Dxx] 26D10: Inequalities involving derivatives and differential and integral operators

Concentration of measure random permutation symmetric group slices of the discrete cube transport inequalities Ewens distribution Chinese restaurant process Deviation’s inequalities for configuration functions

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Samson, Paul-Marie. Transport-entropy inequalities on locally acting groups of permutations. Electron. J. Probab. 22 (2017), paper no. 62, 33 pp. doi:10.1214/17-EJP54. https://projecteuclid.org/euclid.ejp/1502244025

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