 0539 Marek Biskup, Lincoln Chayes, Nicholas Crawford
 Meanfield driven firstorder phase transitions in systems with longrange interactions
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Jan 30, 05

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Abstract. We consider a class of spin systems on $\Z^d$ with vector valued spins $(S_x)$ that interact via the pairpotentials $J_{x,y}S_x\cdot S_y$. The interactions are generally spreadout in the sense that the $J_{x,y}$'s exhibit either exponential or powerlaw falloff. Under the technical condition of reflection positivity and for sufficiently spread out interactions, we prove that the model exhibits a firstorder phase transition whenever the associated meanfield theory signals such a transition. As a consequence, e.g., in dimensions $d\ge3$, we can finally provide examples of the 3state Potts model with spreadout, exponentially decaying interactions, which undergoes a firstorder phase transition as the temperature varies. Similar transitions are established in dimensions $d=1,2$ for powerlaw decaying interactions and in high dimensions for nextnearest neighbor couplings. In addition, we also investigate the limit of infinitely spreadout interactions. Specifically, we show that once the meanfield theory is in a unique "state," then in any sequence of translationinvariant Gibbs states various observables converge to their meanfield values and the states themselves converge to product measure.
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