Percolation on Smooth Gaussian Fields

University of Oxford Junior Applied Maths Seminar- October 2023

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? Level-set percolation on Gaussian fields is a continuum analogue of Bernoulli percolation on graphs. It is believed that many planar Gaussian fields fall into the same universality class as Bernoulli percolation, and a lot of research in the last few years has been devoted to generalising percolation results the the Gaussian case in support of this conjecture.

In this talk, we give a high-level introduction to the theory of Gaussian level-set percolation. We will start by first reviewing the classical theory and some of its key results. Then, we will give some examples of the types of smooth Gaussian fields expected to fall in the Bernoulli universality class, and describe the current progress made in this area . We focus on one result in particular-- the arm-separation lemma-- and briefly outline the tools and methods needed to generalise this result.