What is it about?

A key result to understand percolation on \(\mathbb{Z}^d\) is the sharpness of the phase transition, which is a pair of statements. Subcritical sharpness says that, whenever \(p< p_c\), the probability that two vertices are in the same cluster decays exponentially fast in their distance. On the other side, supercritical sharpness states that, whenever \(p> p_c\), the probability that two vertices are in a same finite cluster decays exponentially fast. The first statement is well-known for all homogeneous graphs. In this work, we prove that the second result does not only hold for the cubic lattice \(\mathbb{Z}^d\) but more generally for any homogeneous graph of polynomial growth. For \(\mathbb{Z}^d\), this yields a new proof of the Grimmett–Marstrand Theorem.

Duke Mathematical Journal, 173(4):745–806, 2024


This work has been used in this companion paper to prove Schramm's Locality Conjecture for the case of graphs with polynomial growth. Philip Easo and Tom Hutchcroft have then used our techniques in combination with other ideas to fully solve this conjecture: their preprint is here. A non-technical presentation of all this can be read in this nice article from Quanta Magazine.


Above is a joint talk Daniel Contreras and I gave about this work at the webinar Percolation Today.