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

Abstract

We prove Griffiths inequalities for the $O(N)$-spin model with inhomogeneous coupling constants and external magnetic field for any $N\ge 2$. 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.

The is an issue with the bijection described in the proof, it is not truly a bijection