Continuity of the phase transition for long-range Bernoulli percolation with exponential decay on $\mathbb{Z}^{2}$

Abstract

We prove continuity of the phase transition for long-range Bernoulli percolation on $\mathbb{Z}^{2}$ with connection probabilities that decay at least exponentially fast. This extends earlier work by Duminil-Copin, Sidoravicius, and Tassion on slabs. Our proof establishes a finite-size criterion that allows us to run a renormalisation argument. As a corollary, we obtain supercritical sharpness, in the sense that for every $β>β_c$, there exists a unique macroscopic cluster in large boxes with high probability.