arXiv subjects

Raphaël Bouyrie

Publications and source records attributed to Raphaël Bouyrie.

2 recordsLinked to original sources

arXiv2017

On quantitative noise stability and influences for discrete and continuous models

Keller and Kindler recently established a quantitative version of the famous Benjamini~--Kalai--Schramm Theorem on noise sensitivity of Boolean functions. The result was extended to the continuous Gaussian setting by Keller, Mossel and Sen by means of a Central Limit Theorem argument. In this work, we present an unified approach of these results, both in discrete and continuous settings. The proof relies on semigroup decompositions together with a suitable cut-off argument allowing for the efficient use of the classical hypercontractivity tool behind these results. It extends to further models of interest such as families of log-concave measures and Cayley and Schreier graphs. In particular we obtain a quantitative version of the B-K-S Theorem for the slices of the Boolean cube.

math.PR↗

arXiv2017

An unified approach to the Junta theorem for discrete and continuous models

In a recent paper, T. Austin has proved an analogous theorem for the continuous torus of the original Junta theorem proved by Friedgut in the case of the Boolean cube. Analogous statements have been established recently in discrete cases such as the discrete Tori by Ellis et.al., and in the case of slices of the Boolean cube by Wimmer and Filmus. In the continuous case, through the notion of geometric influences, a statement has also been established by Keller, Mossel and Sen for Boltzmann probability measures. In this article, we broaden the scope of the arguments of T. Austin to get an unified proof of these results, removing the restriction to Boolean functions. Indeed, the proof of T. Austin relies on semigroup arguments and can be performed in a general framework that covers both Cayley or Schreier graphs or product of log-concave probability measures.

math.PR↗