FKG (and other inequalities) via (generalized) FK representation (and iterated folding)

TL;DR

This paper introduces a novel approach using generalized FK representations and diagrammatic expansions to prove inequalities like FKG, potentially broadening the application of these techniques in statistical mechanics and probability theory.

Contribution

It presents a new method of proving inequalities through iterated foldings and generalized FK representations, expanding the toolkit for analyzing probabilistic inequalities.

Findings

Proves the FKG inequality using generalized FK representation.

Develops a diagrammatic expansion technique for inequalities.

Introduces iterated folding as a tool for probabilistic proofs.

Abstract

In this paper we prove several inequalities by means of diagrammatic expansions, a technique already used in [BG13]. This time we show that iterations of the folding of a probability leads to the proof of some in- equalities by means of a generalized random cluster representation of the iterated foldings. One of the inequalities is the well known FKG inequal- ity, which ends up being proven, quite unexpectedly, by means of the (generalized) FK representation. Although most of the results are not new, we hope that the techniques will find applications in other contexts.