Given a graph with edge weights $\lambda=(\lambda_e)_{e\in E}$, one can define the associated isoperimetric profile $\phi_\lambda(n)$ as the infimum over the total weight of the edge boundaries of vertex subsets of size at least $n$. Our main result shows that if $\phi_\lambda$ grows sufficiently fast (in particular, as any power law), then percolation occurs when each edge is open with probability $p_e=1-e^{-\lambda_e}$. As applications of this result, we prove a conjecture of Sidoravicius, Surgailis and Vares on the existence of percolating truncation for non-summable long-range percolation on $\mathbb{Z}^d$, as well as a conjecture of Easo and Hutchcroft on the asymptotic percolation threshold of transitive graphs with large degree.
As I will explain in the talk, the homogeneous case of this result was recently obtained via a standard Peierls argument together with a clever probabilistic method for counting cutsets, which has no chance to work when weights can be arbitrarily small. Our main contribution is to develop a new Peierls argument which takes into account internal and external connectivity costs in addition to the cost of the blocking surface.
Based on joint work with Ivailo Hartarsky and Augusto Teixeira.