  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.
