Geometric influences II: Correlation inequalities and noise sensitivity

Keller, Nathan; Mossel, Elchanan; Sen, Arnab

Annales de l'I.H.P. Probabilités et statistiques, Tome 50 (2014) no. 4, pp. 1121-1139.

Dans un papier récent, nous avons présenté une nouvelle définition de l'influence dans des produits d'espaces de fonctions continues et montré que des résultats analogues aux résultats les plus importants sur les influences discrètes, comme le théorème KKL, sont valables pour la nouvelle définition dans des espaces gaussiens. Dans cet article, nous prouvons des analogues gaussiens de deux des applications principales des influences : la borne inférieure de Talagrand sur la corrélation de sous-ensembles croissants du cube discret et le théorème de Benjamini-Kalai-Schramm (BKS) sur la sensibilité au bruit. Ensuite nous utilisons les résultats gaussiens pour obtenir des analogues de la borne de Talagrand pour tous les espaces de probabilités discrets et pour retrouver l'analogue du théorème BKS pour des espaces produits biaisés à deux points.

In a recent paper, we presented a new definition of influences in product spaces of continuous distributions, and showed that analogues of the most fundamental results on discrete influences, such as the KKL theorem, hold for the new definition in Gaussian space. In this paper we prove Gaussian analogues of two of the central applications of influences: Talagrand's lower bound on the correlation of increasing subsets of the discrete cube, and the Benjamini-Kalai-Schramm (BKS) noise sensitivity theorem. We then use the Gaussian results to obtain analogues of Talagrand's bound for all discrete probability spaces and to reestablish analogues of the BKS theorem for biased two-point product spaces.

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