Introduction to Percolation: The Bernoulli and the Gaussian

Queen Mary Internal Postgraduate Seminar- February 2024

Imagine driving around Manhattan, where each road segment between two intersections is randomly open or closed, independently and with constant probability. What is the probability that you can drive to infinitely many intersections via open roads? Now, consider a random landscape, where everything below height 0 is submerged underwater, and everything above is traversable land. What is the probability that this random landscape has an infinitely large island?

The connections between these two models-- Bernoulli percolation and Gaussian percolation-- has been a subject of increasing interest in recent times. Similar in many respects, it is conjectured that these models belong to the same ‘universality class,’ but proving this is notoriously difficult. In this talk, we will give a brief introduction to the area of percolation. We will review the history of the subject, explore the link between these two models, and outline the progress that has been made to connect them. Along the way, we hope to introduce some methods and techniques used in the field, and discuss open questions and potential future work.

This talk is introductory and requires no prerequisite knowledge.