Approximation on slabs and uniqueness for inhomogeneous percolation with a plane of defects

2020/10/13 by de Lima, Bernardo N. B., Martineau, Sébastien, Sanna, Humberto C. +1
#60K35 #82B43 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)

paper · doi:10.48550/arxiv.2010.06736

Abstract

Let \mathbbL^d = ( ℤ^d,𝔼^d ) be the d -dimensional hypercubic lattice. We consider a model of inhomogeneous Bernoulli percolation on \mathbbL^d in which every edge inside the s -dimensional hyperplane ℤ^s × \ 0 \^d-s , 2 ≤ 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 ≠ 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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