The 1971 Fortuin-Kasteleyn-Ginibre (FKG) correlation inequality for two monotone functions on a distributive lattice is well known and has seen many applications in statistical mechanics, combinatorics, statistics, probability, and other fields of mathematics. In 2008 the speaker conjectured an extended version of this inequality for all $n>2$ monotone functions on a distributive lattice. This reveals an intriguing connection with the representation theory of the symmetric group. The FKG inequality and its generalization can also be formulated for non-discrete distributive lattices, such as the unit square in ${\mathbb R}^k$ equipped with Lebesgue measure. In this talk we will describe a proof of the $n$ function conjecture in two special cases: (1) for monotone functions on the unit square in ${\mathbb R}^k$ whose (upper) level sets are $k$-dimensional rectangles, and, more significantly, (2) for arbitrary monotone functions on the unit square in ${\mathbb R}^2$. The general case for ${\mathbb R}^k, k>2$ remains open.

This is joint work with Elliott Lieb:  https://arxiv.org/abs/2107.09838