Graduate Student Seminar

The FK-percolation or random-cluster model is a two-parameter family (denote FK(p,q)) of random graphs (specifically, random subgraphs of a fixed graph) and is intimately related to fundamental models in probability and statistical physics, including Bernoulli percolation, the Ising and Potts models, and uniform spanning forests (and trees). In this talk I will introduce the random-cluster model and explain these connections, assuming no prior background in probability or statistical mechanics. A central tool in the study of the model when q≥1 is the FKG inequality, which is a correlation inequality and plays an important role in understanding phase transitions and geometry in this regime. When q<1, this inequality fails, and the behavior of the model becomes much less understood. If time permits, I will describe recent joint work with Vincent Beffara and Corentin Faipeur in which we make some toward understanding this q<1 regime.