  In this paper we establish some relations between percolation on a given
graph G and its geometry. Our main result shows that, if G has polynomial
growth and satisfies what we call the local isoperimetric inequality of
dimension d > 1, then p_c(G) < 1. This gives a partial answer to a question of
Benjamini and Schramm. As a consequence of this result we derive, under the
additional condition of bounded degree, that these graphs also undergo a
non-trivial phase transition for the Ising-Model, the Widom-Rowlinson model and
the beach model. Our techniques are also applied to dependent percolation
processes with long range correlations. We provide results on the uniqueness of
the infinite percolation cluster and quantitative estimates on the size of
finite components. Finally we leave some remarks and questions that arise
naturally from this work.
