On the extension of the FKG inequality to $n$ functions

TL;DR

This paper extends the classical FKG inequality to multiple functions, proving the conjecture in specific cases involving monotone functions on the unit square, with the general higher-dimensional case remaining open.

Contribution

It proves the extended FKG inequality for certain cases in two dimensions, advancing understanding of inequalities for multiple monotone functions.

Findings

Proved the conjecture for monotone functions on the unit square in ${f R}^2$.

Established the inequality for functions with rectangular upper level sets in ${f R}^k$.

The general case for higher dimensions remains unresolved.

Abstract

The 1971 Fortuin-Kasteleyn-Ginibre (FKG) inequality for two monotone functions on a distributive lattice is well known and has seen many applications in statistical mechanics and other fields of mathematics. In 2008 one of us (Sahi) conjectured an extended version of this inequality for all $n>2$ monotone functions on a distributive lattice. Here we prove the conjecture for two special cases: for monotone functions on the unit square in ${\mathbb R}^k$ whose upper level sets are $k$-dimensional rectangles, and, more significantly, for arbitrary monotone functions on the unit square in ${\mathbb R}^2$. The general case for ${\mathbb R}^k, k>2$ remains open.