The Incipient Infinite Cluster for Gaussian Level Set Percolation

CDT Workshop- February 2025

Bernoulli percolation is a statistical physics model, first introduced in the 1960s to model the behaviour of a porous medium. This model exhibits a ‘phase transition,’ where its properties change suddenly as a ‘percolation parameter’ crosses a critical value.

The behaviour of the model at the critical value is of great interest to those in the field, and exhibits many peculiarities. For example, connected components in this model can get arbitrarily large, but not infinitely large. Even though the event that there exists an infinitely large component has probability zero, we can formally ‘condition’ on it, by conditioning on a sequence of events which approximate this null event, and taking limits. In the 80’s, Kesten showed that the two most ‘sensible’ approximation schemes give well-defined limits, and moreover that these limits agree. This gives rise to a new model– the ‘incipient infinite cluster model’-- which almost-surely contains an infinite cluster, but otherwise behaves like critical Bernoulli percolation.

In this talk, we show that an analogue of this result holds for planar Gaussian level-set percolation, a continuous model which serves as a generalisation to Bernoulli percolation.  We will introduce both models, and the background and motivation behind the problem. If time permits, we will discuss how the original proof from the 80s is adapted to the Gaussian setting, by making use of a ‘white noise’ representation for Gaussian fields.

This talk is based on upcoming joint work with Prof. Dmitry Beliaev