Colloquium Mathematics - H. Sanna PhD University Groningen

Title: Two results on inhomogeneous Bernoulli percolation on the d-dimension lattice with a sublattice of defects

Abstract:

In this talk we investigate some aspects 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. Joint work with B. N. B. de Lima and D. Valesin.