with Christoforos Panagiotis
What is it about?
The Locality Theorem states that, under reasonable assumptions, the value of the critical parameter of a graph can be determined with good accuracy if we know a large enough ball of this graph. We argue that the local paradigm can be extended well beyond this specific question. In the setup of homogeneous graphs of dimension \(d\in(1,\infty)\), we prove that the knowledge of a large enough ball suffices to get a fine understanding of the supercritical regime. More precisely, it suffices to obtain bounds yielding supercritical sharpness, isoperimetric bounds on the size of finite clusters, and good knowledge of the analytic continuation of the \(\theta\) map. The first two items correspond to making this previous work locally uniform in the graph. The last item was previously known only for \(\mathbb{Z}^d\). We extend this result to homogeneous graphs of dimension \(d\in(1,\infty)\), and do so in a way that furthermore yields locally uniform estimates. Some reformulation of the \(\theta(p_c)=0\) conjecture is also discussed. In this paper, studying the connectivity of minimal cutsets plays a prominent role; in particular, we answer a question asked by Babson and Benjamini in 1999.