Quantitative Correlation Inequalities via Semigroup Interpolation

Abstract

Most correlation inequalities for high-dimensional functions in the literature, such as the Fortuin-Kasteleyn-Ginibre inequality and the celebrated Gaussian Correlation Inequality of Royen, are qualitative statements which establish that any two functions of a certain type have non-negative correlation. 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 near-optimal quantitative correlation inequalities, including: 
- A {quantitative} version of Royen’s celebrated Gaussian Correlation Inequality [Royen, 2014]. In [Royen, 2014] Royen confirmed a conjecture, open for 40 years, stating that any two symmetric convex sets must be non-negatively correlated under any centered Gaussian distribution. We give a lower bound on the correlation in terms of the vector of degree-2 Hermite coefficients of the two convex sets, conceptually similar to Talagrand’s quantitative correlation bound for monotone Boolean functions over {0,1}ⁿ [M. Talagrand, 1996]. We show that our quantitative version of Royen’s theorem is within a logarithmic factor of being optimal.
- A quantitative version of the well-known FKG inequality for monotone functions over any finite product probability space. This is a broad generalization of Talagrand’s quantitative correlation bound for functions from {0,1}ⁿ to {0,1} under the uniform distribution [M. Talagrand, 1996]; the only prior generalization of which we are aware is due to Keller [Nathan Keller, 2012; Keller, 2008; Nathan Keller, 2009], which extended [M. Talagrand, 1996] to product distributions over {0,1}ⁿ. In the special case of p-biased distributions over {0,1}ⁿ that was considered by Keller, our new bound essentially saves a factor of p log(1/p) over the quantitative bounds given in [Nathan Keller, 2012; Keller, 2008; Nathan Keller, 2009]. We also give {a quantitative version of} the FKG inequality for monotone functions over the continuous domain [0,1]ⁿ, answering a question of Keller [Nathan Keller, 2009].