Central limit theorems in Random cluster and Potts Models

Details Central limit theorems in Random cluster and Potts Models Central limit theorems in Random cluster and Potts Models

2003-08-20

arxiv.org/abs/math/0308190v1

Mathematical Physics Electronic Journal 11 (2005) paper 4 (27 pages)

Mathematics

Probability

version du 19 aout 2003

Scientific paper

We prove that for q>=1, there exists r(q)<1 such that for p>r(q), the number of points in large boxes which belongs to the infinite cluster has a normal central limit behaviour under the random cluster measure phi_{p,q} on Z^d, d>=2. Particularly, we can take r(q)=p_g^* for d=2, which is commonly conjectured to be equal to p_c. These results are used to prove a q-dimensional central limit theorems relative to the fluctuation of the empirical measures for the ground Gibbs measures of the q-state Potts model at very low temperature and the Gibbs measures which reside in the convex hull of them. A similar central limit theorem is also given in the high temperature regime. Some particular properties of the Ising model are also discussed.

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