On the Truncation of Systems with Non-Summable Interactions

Abstract

In this note we consider long range $q$-states Potts models on ${\mathbf{Z}}^d$, $d\ge 2$. For various families of non-summable ferromagnetic pair potentials $\varphi (x)\ge 0$, we show that there exists, for all inverse temperature $\beta >0$, 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 $\varphi (x)\sim \Vert x{\Vert }^{-\alpha }$, $0\le \alpha \le d$. These results are obtained using simple percolation arguments.

18 pages, 4 figures