
Hypercontractivity on the symmetric group
The hypercontractive inequality is a fundamental result in analysis, wit...
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Convex Influences
We introduce a new notion of influence for symmetric convex sets over Ga...
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On Submodular Prophet Inequalities and Correlation Gap
Prophet inequalities and secretary problems have been extensively studie...
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Concentration on the Boolean hypercube via pathwise stochastic analysis
We develop a new technique for proving concentration inequalities which ...
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KruskalKatona for convex sets, with applications
The wellknown KruskalKatona theorem in combinatorics says that (under ...
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A refined determinantal inequality for correlation matrices
Olkin [3] obtained a neat upper bound for the determinant of a correlati...
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Generalized semimodularity: order statistics
A notion of generalized nsemimodularity is introduced, which extends th...
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Quantitative Correlation Inequalities via Semigroup Interpolation
Most correlation inequalities for highdimensional functions in the literature, such as the FortuinKasteleynGinibre (FKG) inequality and the celebrated Gaussian Correlation Inequality of Royen, are qualitative statements which establish that any two functions of a certain type have nonnegative correlation. In this work we give a general approach that can be used to bootstrap many qualitative correlation inequalities for functions over product spaces into quantitative statements. The approach combines a new extremal result about power series, proved using complex analysis, with harmonic analysis of functions over product spaces. We instantiate this general approach in several different concrete settings to obtain a range of new and nearoptimal quantitative correlation inequalities, including: ∙ A quantitative version of Royen's celebrated Gaussian Correlation Inequality. Royen (2014) confirmed a conjecture, open for 40 years, stating that any two symmetric, convex sets must be nonnegatively correlated under any centered Gaussian distribution. We give a lower bound on the correlation in terms of the vector of degree2 Hermite coefficients of the two convex sets, analogous to the correlation bound for monotone Boolean functions over {0,1}^n obtained by Talagrand (1996). ∙ A quantitative version of the wellknown FKG inequality for monotone functions over any finite product probability space, generalizing the quantitative correlation bound for monotone Boolean functions over {0,1}^n obtained by Talagrand (1996). The only prior generalization of which we are aware is due to Keller (2008, 2009, 2012), which extended Talagrand's result to product distributions over {0,1}^n. We also give two different quantitative versions of the FKG inequality for monotone functions over the continuous domain [0,1]^n, answering a question of Keller (2009).
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