Slab Percolation and Phase Transitions for the Ising Model

We prove, using the random-cluster model, a strict inequality between site percolation and magnetization in the region of phase transition for the d-dimensional Ising model, thus improving a result of [CNPR76]. We extend this result also at the case of two plane lattices Z^2x{0,1} (slabs) and give a characterization of phase transition in this case. The general case of N slabs, with N an arbitrary positive integer, is partially solved and it is used to show that this characterization holds in the case of three slabs with periodic boundary conditions. However in this case we do not obtain useful inequalities between magnetization and percolation probability.