Weighted isoperimetry implies percolation

Abstract

Consider an infinite edge-weighted graph satisfying an isoperimetric inequality of the type $\|\partial A\|\geq C|A|^\alpha$ for some $\alpha,C>0$, where $\|\partial A\|$ denotes the weighted size of the edge boundary of $A$. We prove that, for $C$ large enough depending on $\alpha$, if each edge is open independently with probability given by its weight, then any vertex is connected to infinity with positive probability. The result also holds under weaker isoperimetric assumptions and on finite graphs. The proof brings a new perspective on the recent proof of the Benjamini--Schramm conjecture concerning the same problem with homogeneous weights. The crucial novelty in our proof is that, rather than simply counting cutsets, we introduce a new Peierls argument which takes into account internal and external connectivity costs in addition to the cost of the blocking surface. We provide two applications for the above result. First, we show that every non-summable long-range percolation on $\mathbb{Z}^d$, $d\geq 2$, admits a percolating truncation, solving a conjecture of Sidoravicius, Surgailis and Vares and its generalization by Friedli and de Lima. Secondly, we show that there exists a universal constant $C<\infty$ such that $p_{\mathrm{c}} \leq C/\Delta$ for every transitive graph of superlinear growth and vertex degree $\Delta$, thus proving a conjecture of Easo and Hutchcroft.