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 5]. We extend this result also at the case of two plane lattices (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. AMS classification: 60K35, 82B20, 82A25