On the Truncation of Systems with Non-Summable Interactions

Abstract

In this note we consider long range $q$-states Potts models on $\mathbf{Z}^d$, $d\geq 2$. For various families of non-summable ferromagnetic pair potentials $ϕ(x)\geq 0$, we show that there exists, for all inverse temperature $β>0$, an integer $N$ such that the truncated model, in which all interactions between spins at distance larger than $N$ are suppressed, has at least $q$ distinct infinite-volume Gibbs states. This holds, in particular, for all potentials whose asymptotic behaviour is of the type $ϕ(x)\sim \|x\|^{-α}$, $0\leqα\leq d$. These results are obtained using simple percolation arguments.