ÍâÍøÌìÌÃ

In this talk, we prove that Bernoulli percolation on graphs withÌýisoperimetric dimension $d>4$ undergoes a non-trivial phase transition (in theÌýsense that $p_c<1$). As a corollary, we obtain that the critical point ofÌýBernoulli percolation on infinite quasi-transitive graphs (in particular,ÌýCayley graphs) with super-linear growth is strictly smaller than $1$, thusÌýanswering a conjecture of Benjamini and Schramm from 1996. The proof relieson a new technique consisting in expressing certain functionals of theÌýGaussian Free Field (GFF) in terms of connectivity probabilities for aÌýpercolation model in a random environment. Then, we integrate out theÌýrandomness in the edge-parameters using a multi-scale decomposition of theÌýGFF. We believe that a similar strategy could lead to proofs of theÌýexistence of a phase transition for various other models.