Locality of percolation for graphs with polynomial growth

We prove Schramm's Locality Conjecture under the assumption of polynomial growth. In other words, if a homogeneous graph with polynomial growth satisfies \(p_c < 1\), it is enough to know how a large ball looks like to be able to compute \(p_c\) up to epsilon. This work builds on two recent works regarding homogeneous graphs of polynomial growth: the structure theorem of Tessera–Tointon and our description of supercritical percolation.

Electronic Communications in Probability, 28:1–9, 2023

Philip Easo and Tom Hutchcroft have then used our techniques in combination with other ideas to fully solve Schramm's Locality Conjecture: their preprint is here. A non-technical presentation of all this can be read in this nice article from Quanta Magazine.