Percolation in Gaussian fields

It has been conjectured that the excursion sets of stationary Gaussian fields behave like Bernoulli percolation. In particular, it has been shown that, under certain assumptions on the covariance kernel, they exhibit a phase transition: below a certain critical level, the excursion set almost surely contains a unique unbounded component, and above it all components are bounded. In the two-dimensional case, under very mild assumptions, the critical level is known to be 0. It is also known under some assumptions, and is believed to be true more generally, that there are no infinite clusters at the critical level. Following ideas from Bernoulli percolation, we show that there is a natural way to define a field conditioned to have an unbounded excursion set at the critical level. As in the case of Bernoulli percolation, we call this the Incipient Infinite Cluster (IIC).

This is joint work with Julius Villar.