UBC Probability Seminar: Johannes Baeumler

For long-range percolation on ℤ with translation-invariant edge kernel J, it is a classical theorem of Aizenman and Newman (1986) that the phase transition is discontinuous when J(x−y) is of order |x−y|−2 and that there is no phase transition at all when J(x−y)=o(|x−y|−2). We prove analogous theorems for the hierarchical lattice, where the relevant threshold is at |x−y|−2dloglog|x−y| rather than |x−y|−2: There is a continuous phase transition for kernels of larger order, a discontinuous phase transition for kernels of exactly this order, and no phase transition at all for kernels of smaller order. Based on joint work with Tom Hutchcroft.