UVictoria Probability and Dynamics Seminar: Nathan Zelesko

Site percolation models are probability distributions of 2-colorings of the vertices of locally finite infinite graphs. Historically, they have been studied on graphs exhibiting symmetries such as (quasi)-transitivity. In joint work with Alexander Glazman and Matan Harel, we instead only assume that the graph is planar. We show that a large class of site percolation models on any planar graph contains either zero or infinitely many infinite connected components. This includes the case of Bernoulli percolation at parameter p \leq 1/2, resolving a conjecture from the work of Benjamini and Schramm from 1996. In this talk, I will discuss the main ingredients of the proof, with emphasis on the properties we require of the model.