I will introduce the "mass-transport method" and give some applications to percolation. My talk will largely be based on a paper of Olle Haggstrom called "Invariant percolation on trees and the mass-transport method." If time permits, I might also sketch an application to a problem in first-passage percolation.

In this sequel to the talk I gave last semester, join me as we generalize the central limit theorem beyond the bounds of finite mean and variance. Recoil in horror at the 4 parameter family of random variables that we discover, stable in form and terrible in characteristic function. Do not gaze too long into their basins of attraction, lest the stable random variables gaze back into you! If time permits, I will give some examples where stable random variables arise as limits.

I plan on introducing the model, the critical parameters and giving a few of the results on parallels to discrete percolation.

In this talk I will report on the 2017 paper "Sharp phase transition for the random-cluster and Potts models via decision trees", by Hugo Duminil-Copin, Aran Raoulfi and Vincent Tassion. The paper gives a short proof of the following facts: in subcritical percolation, the probability of crossing a box is exponentially small in the size of the box; in supercritical percolation, the probability that an infinite cluster exists is superlinear in the percolation parameter. Their proof uses the concept of "decision tree", that comes from computer science, to set up a differential inequality for the box-crossing probabilites. From the differential inequality, they are able to finish the proof in a very clever way, using nothing but calculus. The talk will present their result and the structure of the proof in the special case of Bernoulli percolation . After that, it will explain in detail how they go from the differential inequality to the sharpness theorem. The goal is to understand what made their argument so effective.

Abstract Here

Abstract Here