  In this paper, we prove that Bernoulli percolation on bounded degree graphs
with isoperimetric dimension $d>4$ undergoes a non-trivial phase transition (in
the sense that $p_c<1$). As a corollary, we obtain that the critical point of
Bernoulli percolation on infinite quasi-transitive graphs (in particular,
Cayley graphs) with super-linear growth is strictly smaller than 1, thus
answering a conjecture of Benjamini and Schramm. The proof relies on a new
technique consisting in expressing certain functionals of the Gaussian Free
Field (GFF) in terms of connectivity probabilities for percolation model in a
random environment. Then, we integrate out the randomness in the
edge-parameters using a multi-scale decomposition of the GFF. We believe that a
similar strategy could lead to proofs of the existence of a phase transition
for various other models.
