Central Limit Theorems in Random Cluster and Potts Models

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 φp,q on Z, d ≥ 2. Particularly, we can take r(q) = p∗g for d = 2, which is commonly conjectured to be equal to pc. These results are used to prove a qdimensional 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.