Correlation Inequalities for Generalized Potts Model: General Griffiths' Inequalities

TL;DR

This paper extends correlation inequalities from the Ising model to the generalized q-Potts model, establishing non-negativity of certain expectations and covariances for a broad class of interactions.

Contribution

It introduces correlation inequalities for the generalized Potts model with arbitrary spin interactions, generalizing known results from the Ising model.

Findings

Proved that the expectation of products of spins is non-negative.

Established that the covariance between spin products is non-negative.

Extended Griffiths' inequalities to the q-Potts model with generalized interactions.

Abstract

In this paper, correlation inequalities which have been considered on Ising model are extended to q-Potts model. It is considered on generalized Potts model with interaction of any number of spins. We replace the set of spin values $F=\{1,2,..., q\}$ by the centered set $F=\{-(q-1)/2,-(q-3)/2,... ,(q-3)/2,(q-1)/2\}$. Let $N$ be the subset of one-dimensional lattice with $n$ vertices, $\g=(\s_1,\s_2,...,\s_n):N \to F^c$ be a configuration where ${(\s_i)}_\g$ is the number which appears as the ith spin (component) in $\g$ and $\s_i$ be a random variable whose value at $\g$ is ${(\s_i)}_\g$. Define $\s^R=\prod_{i \in R}\s_i$ for any list $R$ where any $i \in R$ implies that $i \in N$. We first prove that $<\s^R > \ge 0$ then we prove that for any two lists $R$ and $S$, we have $<\s^R \s^S >- < \s^R > < \s^S > \ge 0$.