Royen's proof of the Gaussian correlation inequality as a supersymmetric dimensional reduction

Abstract

We revisit Royen's proof of the Gaussian correlation inequality from a supersymmetric point of view. Many key elements in Royen's proof of this inequality have natural geometric interpretations in terms of supersymmetric dimensional reduction from ${\mathbb{R}}^{3|2}$ to ${\mathbb{R}}^{1|0}$. In particular, the auxiliary multivariate Gamma distributions appearing in Royen's Laplace-transform argument arise naturally as the body of a supersymmetric radial variable on ${\mathbb{R}}^{3|2}$. The generalization to the half-integer multivariate Gamma case also follows naturally as a dimensional reduction from ${\mathbb{R}}^{k+2|2}$ to ${\mathbb{R}}^{k|0}$, and we derive an exact interpolation identity for general smooth test functions. This provides an example in which the supersymmetric localization method is applied to prove correlation inequalities with continuous interpolation.

16 pages