On Reverse Hypercontractivity

Elchanan Mossel, Krzysztof Oleszkiewicz, Arnab Sen

Research output: Contribution to journalArticlepeer-review

49 Scopus citations

Abstract

We study the notion of reverse hypercontractivity. We show that reverse hypercontractive inequalities are implied by standard hypercontractive inequalities as well as by the modified log-Sobolev inequality. Our proof is based on a new comparison lemma for Dirichlet forms and an extension of the Stroock-Varopoulos inequality. A consequence of our analysis is that all simple operators L = Id - E as well as their tensors satisfy uniform reverse hypercontractive inequalities. That is, for all q < p < 1 and every positive valued function f for t ≥ we have {pipe}{pipe}e-tLf{pipe}{pipe}q ≥ {pipe}{pipe}f{pipe}{pipe}p. This should be contrasted with the case of hypercontractive inequalities for simple operators where t is known to depend not only on p and q but also on the underlying space. The new reverse hypercontractive inequalities established here imply new mixing and isoperimetric results for short random walks in product spaces, for certain card-shufflings, for Glauber dynamics in high-temperatures spin systems as well as for queueing processes. The inequalities further imply a quantitative Arrow impossibility theorem for general product distributions and inverse polynomial bounds in the number of players for the non-interactive correlation distillation problem with m-sided dice.

Original languageEnglish (US)
Pages (from-to)1062-1097
Number of pages36
JournalGeometric and Functional Analysis
Volume23
Issue number3
DOIs
StatePublished - Jun 2013

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