On the Truncation of Systems with Non-Summable Interactions

Details On the Truncation of Systems with Non-Summable Interactions On the Truncation of Systems with Non-Summable Interactions

2005-04-28

arxiv.org/abs/cond-mat/0504737v1

Physics

Condensed Matter

Statistical Mechanics

18 pages, 4 figures

Scientific paper

10.1007/s10955-005-8023-9

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 $\phi(x)\geq 0$, 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 $\phi(x)\sim \|x\|^{-\alpha}$, $0\leq\alpha\leq d$. These results are obtained using simple percolation arguments.

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