Correlation Inequalities for Generalized Potts Model: General Griffiths' Inequalities

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 $F=\{1,2,\mathrm{.}\mathrm{.}\mathrm{.},q\}$ by the centered set $F={-(q-1)/2,-(q-3)/2,… ,(q-3)/2,(q-1)/2}$\mathrm{.}Let$N$bethesubsetofone-dimensionallatticewith$n$vertices,$\g=(\s_1,\s_2,…,\s_n):N \to F^c$beaconfigurationwhere${(\s_i)}\g$isthenumberwhichappearsastheithspin(component)in$\g$and$\s_i$bearandomvariablewhosevalueat$\g$is${(\s_i)}\g$\mathrm{.}Define$\s^R=\prod_{i \in R}\s_i$foranylist$R$whereany$i \in R$impliesthat$i \in N$\mathrm{.}Wefirstprovethat$<\s^R > \ge 0$ then we prove that for any two lists $R$ and $S$, we have $<{\text{s}}^R{\text{s}}^S>-<{\text{s}}^R><{\text{s}}^S>\ge 0$.