Continuity of the phase transition for long-range Bernoulli percolation with exponential decay on Z²

We prove continuity of the phase transition for long-range Bernoulli percolation on Z² 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>βc, there exists a unique macroscopic cluster in large boxes with high probability.