Topics in inhomogeneous Bernoulli percolation: A study of two models

This thesis is an investigation of some aspects of inhomogeneous Bernoulli bond percolation in two different graphs.

First, we show the continuity of the critical curve in the qp-plane for inhomogeneous Bernoulli bond percolation on ladder graphs. We prove that such property can be achieved even if the graph possesses infinitely many columnar inhomogeneities, provided that they are not too close from each other.

Second, we study the model of inhomogeneous Bernoulli bond percolation on the ordinary d-dimensional hypercubic lattice, d higher than 3, with an s-dimensional sublattice of defects, s smaller than d. In this model, every edge inside the s-dimensional sublattice is open with probability q and every other edge is open with probability p. We prove two results: first, the uniqueness of the infinite cluster in the supercritical phase of parameters (p,q), whenever p is different from the threshold for homogeneous percolation; second, we show that, for any p smaller than this threshold, the critical points in the pq-plane can be approximated by critical points of slabs, in the spirit of the classical theorem of Grimmett and Marstrand for homogeneous percolation.