On Reverse Hypercontractivity
详细信息 查看全文
- 作者：Elchanan Mossel (1) (2)
Krzysztof Oleszkiewicz (3)
Arnab Sen (4) - 关键词：Primary ; 60E15 ; Secondary ; 60J27
- 刊名：Geometric And Functional Analysis
- 出版年：2013
- 出版时间：June 2013
- 年：2013
- 卷：23
- 期：3
- 页码：1062-1097
- 全文大小：431KB
- 参考文献：1. N. Alon, U. Maurer and A. Wigderson. Unpublished results, 1991.
2. Arrow J.: A difficulty in the concept of social welfare. The Journal of Political Economy 4(58), 328-46 (1950) CrossRef
3. J. Arrow. / Social choice and individual values, Wiley, Hoboken (1963).
4. N. Alon and J.H. Spencer. / The probabilistic method, 3rd edn. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, Hoboken, NJ (2008) With an appendix on the life and work of Paul Erdós. MR 2437651 (2009j:60004).
5. Bakry D.: L’hypercontractivit’e et son utilisation en th’eorie des semigroupes. Lectures on Probability Theory 1581, 1-14 (1994) CrossRef
6. W. Beckner. Inequalities in Fourier analysis, / Ann. Math. (2) (1)102 (1975), 159-82. MR 0385456 (52 #6317).
7. C. Borell and S. Janson. Converse hypercontractivity, Séminaire Initiation á l’Analyse, 21e année, 1981/82. / Publ. Math. Univ. Pierre Marie Curie, vol. 54, No. 4.
8. S.G. Bobkov and M. Ledoux. On modified logarithmic Sobolev inequalities for Bernoulli and Poisson measures, / J. Funct. Anal. (2)156 (1998), 347-65. MR1636948 (99e:60051).
9. A. Bonami. étude des coefficients de Fourier des fonctions de / L / p ( / G), / Ann. Inst. Fourier (Grenoble) 20 (1970), no. fasc. 2, 335-02 (1971). MR 0283496 (44 #727).
10. C. Borell. Positivity improving operators and hypercontractivity, / Math. Z. (2)180 (1982), 225-34. MR 661699 (84b:47029).
11. S.G. Bobkov and P. Tetali. Modified log-Sobolev inequalities, / mixing and hypercontractivity, Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing (New York). ACM, New York (2003), pp. 287-96 (electronic). MR 2120475 (2005k:60027).
12. S.G. Bobkov and P. Tetali. Modified logarithmic Sobolev inequalities in discrete settings, / J. Theoret. Probab. (2)19 (2006), 289-36. MR 2283379 (2007m:60215).
13. P. Diaconis, J.A. Fill and J. Pitman. Analysis of top to random shuffles, / Combin. Probab. Comput. (2)1 (1992), 135-55. MR 1179244 (93f:60011).
14. P. Diaconis and L. Saloff-Coste. Logarithmic Sobolev inequalities for finite Markov chains, / Ann. Appl. Probab. 6 (3)(1996), 695-50. MR 1410112 (97k:60176).
15. S. Goel. Modified logarithmic Sobolev inequalities for some models of random walk, / Stochastic Process. Appl. (1)114 (2004), 51-9. MR 2094147 (2006h:60122).
16. F. Gao and J. Quastel. Exponential decay of entropy in the random transposition and Bernoulli–Laplace models, / Ann. Appl. Probab. (4)13 (2003), 1591-600. MR 2023890 (2005a:60155).
17. L. Gross. Logarithmic Sobolev inequalities, / Amer. J. Math. (4)97 (1975), 1061-083. MR 0420249 (54 #8263).
18. L. Gross. Logarithmic Sobolev inequalities—a survey. Vector space measures and applications (Proc. Conf., Univ. Dublin, Dublin, 1977), I, / Lecture Notes in Math., vol. 644. Springer, Berlin (1978), pp. 196-03. MR 502406 (80f:46035).
19. S. Janson, Gaussian Hilbert Spaces. Cambridge Tracts in Mathematics vol. 129, / Cambridge University Press, Cambridge (1997). MR 99f:60082
20. M. Jerrum and J. Son. Spectral gap and log-Sobolev constant for balanced matroids, / Proceedings of the 43rd IEEE Symposium on Foundations of Computer Science(FOCS-2) (2002), 721-29.
21. G. Kalai. A Fourier-theoretic perspective on the Concordet paradox and Arrow’s theorem, / Adv. in Appl. Math. (3)29 (2002), 412-26. MR 1942631 (2003h:91046).
22. J. Kahn, G. Kalai and N. Linial. The influence of variables on Boolean functions, / Proceedings of the 29th Annual Symposium on Foundations of Computer Science 1988, pp. 68-0.
23. R. Lata?a and K.Oleszkiewicz. Between Sobolev and Poincaré, Geometric aspects of functional analysis, / Lecture Notes in Math., vol. 1745. Springer, Berlin (2000), pp. 147-68. MR 1796718 (2002b:60025).
24. T. Lee and H. Yau. Logarithmic Sobolev inequality for some models of random walks, / Ann. Probab. (4)26 (1998), 1855-873. MR1675008 (2001b:60090).
25. F. Martinelli and E. Olivieri. Approach to equilibrium of Glauber dynamics in the one phase region. I. The attractive case, / Comm. Math. Phys. (3)161 (1994), 447-86. MR 1269387 (96c:82040).
26. Martinelli. Approach to equilibrium of Glauber dynamics in the one phase region. II. The general case, / Comm. Math. Phys. (3)161 (1994), 487-14. MR 1269388 (96c:82041).
27. E. Mossel and R. O’Donnell. Coin flipping from a cosmic source: on error correction of truly random bits, / Random Structures Algorithms (4)26 (2005), 418-36. MR 2139875 (2006c:60012).
28. E. Mossel, R. O’Donnell, O. Regev, J.E. Steif and B. Sudakov Non-interactive correlation distillation, inhomogeneous Markov chains, and the reverse Bonami-Beckner inequality, / Israel J. Math. 154 (2006), 299-36. MR 2254545 (2008b:60018).
29. Mossel E.: A quantitative Arrow theorem. Probability Theory and Related Fields 1(154), 49-8 (2012) CrossRef
30. E. Nelson. The free Markoff field, / J. Functional Analysis 12 (1973), 211-27. MR0343816 (49 #8556).
31. K. Oleszkiewicz. On a nonsymmetric version of the Khinchine-Kahane inequality, / Stochastic inequalities and applications, Progr. Probab., vol. 56. Birkh?user, Basel (2003), pp. 157-68. MR 2073432 (2005f:60049).
32. L. Saloff-Coste. Lectures on finite Markov chains, Lectures on probability theory and statistics (Saint-Flour, 1996), / Lecture Notes in Math. vol. 1665. Springer, Berlin (1997), pp. 301-13. MR 1490046 (99b:60119).
33. D.W. Stroock. / An introduction to the theory of large deviations, Universitext. Springer, New York (1984). MR 755154 (86h:60067a).
34. D. Stroock and B. Zegarli’nski. The equivalence of the logarithmic Sobolev inequality and the Dobrushin-Shlosman mixing condition, / Comm. Math. Phys. (2)144 (1992), 303-23. MR 1152374 (93b:82005).
35. Stroock. The logarithmic Sobolev inequality for discrete spin systems on a lattice, / Comm. Math. Phys. (1)149 (1992), 175-93. MR 1182416 (93j:82013).
36. M. Talagrand. On Russo’s approximate 0-1 law, / Annals of Probability 22 (1994), 1576-587. MR 1303654 (96g:28009).
37. N. Th. Varopoulos. Hardy–Littlewood theory for semigroups, / J. Funct. Anal. (2)63 (1985), 240-60. MR 803094 (87a:31011).
38. R. Wilson. Social choice theory without the Pareto principle, / Journal of Economic Theory (3)5 (1972), 478-86. MR 0449494 (56 #7796).
39. P.Wolff. Hypercontractivity of simple random variables, / Studia Mathematica 180 (2007), 219-26. MR 2314078 (2008g:60009).
40. L. Wu. A new modified logarithmic Sobolev inequality for Poisson point processes and several applications, / Probab. Theory Related Fields (3)118 (2000), 427-38. MR 1800540 (2002f:60109).
41. K.Yang. On the (im)possibility of non-interactive correlation distillation, LATIN 2004: Theoretical informatics, / Lecture Notes in Comput. Sci., vol. 2976. Springer, Berlin (2004), pp. 222-31. MR 2095197 (2005f:94045).
42. B. Zegarliński. Dobrushin uniqueness theorem and logarithmic Sobolev inequalities, / J. Funct. Anal. (1)105 (1992), 77-11. MR 1156671 (93d:58190). - 作者单位：Elchanan Mossel (1) (2)
Krzysztof Oleszkiewicz (3)
Arnab Sen (4)
1. Statistics and Computer Science, University of California, 367 Evans Hall, Berkeley, CA, USA
2. Faculty of Mathematics and Computer Science, Weizmann Institute, Rehovot, Israel
3. Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097, Warsaw, Poland
4. Statistical Laboratory, DPMMS, Wilberforce Road, Cambridge, CB3 0WB, UK - ISSN：1420-8970
文摘
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 ${L = Id - \mathbb{E}}$ as well as their tensors satisfy uniform reverse hypercontractive inequalities. That is, for all q?<?p?<?1 and every positive valued function f for ${t \geq \log \frac{1-q}{1-p}}$ we have ${\| e^{-tL}f\|_{q} \geq \| f\|_{p}}$ . 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.