Transport-entropy inequalities on locally acting groups of permutations

TL;DR

This paper extends transport-entropy inequalities to locally acting permutation groups, generalizing concentration results and providing new deviation bounds for permutation-related functions.

Contribution

It introduces transport-entropy inequalities for permutation groups acting locally, broadening the scope of concentration results and including measures like Ewens distributions.

Findings

Transport-entropy inequalities hold for various permutation groups.

Concentration properties follow from these inequalities.

New deviation bounds for permutation cycle functions.

Abstract

Following Talagrand's concentration results for permutations picked uniformly at random from a symmetric group [Tal95], Luczak and McDiarmid have generalized it to more general groups G 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 concentra- tion properties are consequences of these inequalities. The results are also gen- eralised 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 [GRST15], that con- tribute 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.