Quantitative correlation inequalities via extremal power series

Many correlation inequalities for high-dimensional functions in the literature, such as the Harris-Kleitman inequality, 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. Previous work has used Markov semigroup arguments to obtain quantitative extensions of some of these correlation inequalities. In this work, we augment this approach with a new extremal bound on power series, proved using tools from complex analysis, to obtain a range of new and near-optimal quantitative correlation inequalities. These new results include: 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) (Talagrand in Combinatorica 16(2):243-258, 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 (Talagrand in Combinatorica 16(2):243-258, 1996). 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 Keller (Eur J Comb 33:1943-1957, 2012; Improved FKG inequality for product measures on the discrete cube, 2008; Influences of variables on Boolean functions. PhD thesis, Hebrew University of Jerusalem, 2009).