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

TL;DR

This paper offers a supersymmetric perspective on Royen's proof of the Gaussian correlation inequality, revealing geometric interpretations and natural generalizations via dimensional reduction.

Contribution

It introduces a supersymmetric framework to interpret Royen's proof and extends the Gaussian correlation inequality to half-integer multivariate Gamma distributions.

Findings

Supersymmetric dimensional reduction explains key elements of Royen's proof.

The approach generalizes the inequality to half-integer multivariate Gamma distributions.

Supersymmetric localization method applies to correlation inequalities with continuous parameters.

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 $\mathbb{R}^{k|0}$. This provides an example in which the supersymmetric localization method is applied to prove correlation inequalities with continuous parameters.