Algumas questões em percolação anisotrópica

Abstract

In this work we study some aspects of anisotropic independent bond percolation on the slab Z2 f0; : : : ; kg and on the lattice Z2. We consider anisotropic independent bond percolation on the slab Z2 f0; : : : ; kg, where we suppose that the vertical bonds are open with probability pv, while the horizontal bonds are open with probability ph. We study the critical curves for these models and establish their continuity and strictmonotonicity. The results can be extended to anisotropic independent bond percolation on Z3. Later, we consider an anisotropic independent bond percolation model on Z2, i.e., let p = (ph; pv) 2 [0; 1]2 with pv > ph and declare each horizontal (respectively vertical) edge of Z2 to be open with probability ph (respectively pv), and otherwise closed, independentlyof all other edges. Let Pp denote the corresponding probability measure. Let x = (x1; x2) 2 Z2 with 0 < x1 < x2, and x0 = (x2; x1) 2 Z2. Its natural to ask how behaves the connectivity function Pp(f0 ! xg), and whether anisotropy in percolation probabilities implies the strict inequality Pp(f0 ! xg) > Pp(f0 ! x0g). We give afirmative answer to this question, at least in the highly supercritical regime.