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تسجيل مستخدم جديدApproximation on slabs and uniqueness for inhomogeneous percolation with a plane of defects
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Let $ mathbb{L}^{d} = ( mathbb{Z}^{d},mathbb{E}^{d} ) $ be the $ d $-dimensional hypercubic lattice. We consider a model of inhomogeneous Bernoulli percolation on $ mathbb{L}^{d} $ in which every edge inside the $ s $-dimensional hyperplane $ mathbb{Z}^{s} times { 0 }^{d-s} $, $ 2 leq s < d $, is open with probability $ q $ and every other edge is open with probability $ p $. We prove the uniqueness of the infinite cluster in the supercritical regime whenever $ p eq p_{c}(d) $, where $ p_{c}(d) $ denotes the threshold for homogeneous percolation, and that the critical point $ (p,q_{c}(p)) $ can be approximated on the phase space by the critical points of slabs, for any $ p < p_{c}(d) $.
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