Griffiths inequalities for the $O(N)$-spin model

TL;DR

This paper proves Griffiths inequalities for the $O(N)$-spin model with inhomogeneous couplings and magnetic field, extending known results to a broader class of models.

Contribution

It introduces a novel representation of $O(N)$-spins via random paths, generalizing the random current approach from the Ising model.

Findings

Established Griffiths inequalities for all $N \\geq 2$

Developed a new representation linking $O(N)$-spins to random paths

Extended the switching lemma analogy to $O(N)$ models

Abstract

We prove Griffiths inequalities for the $O(N)$-spin model with inhomogeneous coupling constants and external magnetic field for any $N\geq 2$. This is achieved by using a representation of $O(N)$-spins in terms of random paths that reduces to the random current representation of the Ising model for $N=1$ and an identity that is analogous to the switching lemma for random currents.