MPEJ Volume 11, No. 4, 27 pp.
Received: Mar 11, 2004. Revised: Sep 18, 2005. Accepted: Oct 14, 2005.
O. Garet
Central limit theorems for the Potts Model
ABSTRACT: We prove various $q$-dimensional Central Limit Theorems for
the occuring of the states in the $q$-state Potts model on $\Zd$ at
inverse temperature $\beta$, provided that $\beta$ is sufficiently far
from the critical point $\beta_c$. When $(d=2)$ and ($q=2$ or $q\ge
26$), the theorems apply for each $\beta\ne\beta_c$. In the uniqueness
region, a classical Gaussian limit is obtained. In the phase
transition regime, the situation is more complex: when $(q\ge 3)$, the
limit may be Gaussian or not, depending on the Gibbs measure which is
considered. Particularly, we show that free boundary conditions lead
to a non-Gaussian limit. Some particular properties of the Ising
model are also discussed. The limits that are obtained are identified
relatively to FK-percolation models.
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