Central limit theorems in Random cluster and Potts Models

Olivier Garet (1)

1 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans

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.

Mots clés: random cluster measure; percolation; coloring model; Central Limit Theorem; Potts Model; Ising Model

Domaines: Probabilités [math.PR]

Olivier Garet. Central limit theorems in Random cluster and Potts Models. Mathematical Physics Electronic Journal, 2005, 11, paper 4 (27 p.). ⟨hal-00000552⟩