On large clusters in percolation

Abstract : This thesis is dedicated to the study of large clusters in percolation and is divided into four articles. Models under consideration are Bernoulli percolation, FK percolation and oriented percolation. Key ideas are renormalization, large deviations, FKG and BK inequalities and mixing properties. We prove a large deviation principle for clusters in the subcritical phase of Bernoulli percolation. We use FKG inequality for the lower bound. As for the upper bound, we use BK inequality together with a skeleton coarse graining. We establish large deviations estimates of surface order for the density of the maximal cluster in a box in dimension two for supercritical FK percolation. We use renormalization and we compare a block process with a site-percolation process whose parameter of retention is close to one. We prove that large finite clusters are distributed accordingly to a Poisson process in supercritical FK percolation and in all dimensions. The proof is based on the Chen-Stein method and it makes use of mixing properties such as the ratio weak mixing property. We establish a large deviation principle of surface order for the supercritical oriented percolation. The framework is that of the non-oriented case, but difficulties arise despite of the Markovian nature of the oriented process. We give new block estimates, which describe the behaviour of the oriented process. We also obtain the exponential decay of connectivities outside the cone of percolation, which is the typical shape of an infinite cluster.
Document type :
Mathematics [math]. Université Paris Sud - Paris XI, 2004. English
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Submitted on : Friday, January 7, 2005 - 12:58:26 PM
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Olivier Couronné. On large clusters in percolation. Mathematics [math]. Université Paris Sud - Paris XI, 2004. English. 〈tel-00007948〉



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