ar X iv : 0 70 5 . 37 81 v 1 [ m at h . PR ] 2 5 M ay 2 00 7 POISSON APPROXIMATION FOR LARGE CLUSTERS IN THE SUPERCRITICAL FK MODEL

Using the Chen-Stein method, we show that the spatial distribution of large finite clusters in the supercritical FK model approximates a Poisson process when the ratio weak mixing property holds. 1. Introduction. We consider here the behaviour of large finite clusters in the supercritical FK model. In dimension two and more, their typical structure is described by the Wulff shape [4, 5, 6, 8, 9, 10, 11]. An interesting issue is the spatial distribution of these large finite clusters. Because of their rarity, a Poisson process naturally comes to mind. Indeed, we prove that the point process of the mass centers of large finite clusters sharply approximates a Poisson process. Furthermore, considering large finite clusters in a large box such that their mean number is not too large, we observe Wulff droplets distributed according to this Poisson process. Redig and Hostad have recently studied the law of large finite clusters in a given box [20]. Their aim was different, in that they obtained accurate estimates on the law of the maximal cluster in the box, but the intermediate steps are similar. In the supercritical regime they considered only Bernoulli percolation and not FK percolation. our main result is based on a second moment inequality. We have to control the interaction between two clusters. For this, we suppose that ratio weak mixing holds [2]. This property allows us to apply the Chen-Stein method in order to get the approximation by a Poisson process. The ratio weak mixing holds in dimension two as soon as dual connectivities are exponentially decreasing [2]. For dimensions at least three, we prove that ratio weak mixing holds for p close enough to 1. Hence our main results are valid in all dimensions for p large enough. The following section is devoted to the statement of our results. In section 3, we define the FK model. We recall the weak and the ratio weak mixing properties and we state a perturbative mixing result in section 4. Section 5 contains the definition