Uniqueness of the Phase Transition
We state and prove the celebrated result that the critical value where the expected cluster size blows equals that where the probability of a vertex being in an infinite component agree. This uniqueness of the phase transition was independently proved by Menshikov and by Aizenman and Barsky. This theorem is the starting point of the investigation of high-dimensional percolation. We start with the recent and beautiful Duminil-Copin and Tassion proof, continuing with the original Aizenman and Barsky proof that relies on differential inequalities for the so-called percolation magnetization. The Aizenman-Barsky differential inequalities also play a pivotal role in the identification of mean-field critical exponents for percolation in high dimensions.