Decorrelation as an avatar of convexity

Abstract

If $X$ is a Polish space then we show that the product measure on $X^{\infty }$ is guaranteed to minimize $c$-energy amongst exchangeable measures with fixed marginals if and only if the interaction kernel $c$ defines a convex energy functional on probability measures. A reformulation of this condition close to the theory of positive definite functions is highlighted.

8 pages. Comments are welcome!