On Reverse Hypercontractivity

Elchanan Mossel, Krzysztof Oleszkiewicz, Arnab Sen

Research output: Contribution to journalArticlepeer-review

36 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

Bibliographical note

Funding Information:
The first author is supported by NSF DMS 0548249 (CAREER) and NSF DMS 1106999 awards, by DOD ONR grant N000141110140, by ISF grant 1300/08 and by a Minerva Grant. Most of this work was conducted when the author was at the Weizmann institute.

Funding Information:
The second author is partially supported by Polish MNiSzW Grant N N201 397437. Arnab Sen is partially supported by EPSRC grant EP/G055068/1.

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