On the Truncation of Systems with Non-Summable Interactions

TL;DR

This paper proves that for long-range Potts models with non-summable interactions, truncating interactions beyond a certain distance still results in multiple Gibbs states, highlighting the persistence of phase coexistence.

Contribution

It demonstrates that truncating non-summable long-range interactions preserves multiple Gibbs states in Potts models, extending understanding of phase behavior in such systems.

Findings

Existence of at least q distinct Gibbs states after truncation

Results apply to potentials with decay rate  

Uses percolation arguments for proofs

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 $\phi(x)\geq 0$, 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.