Stochastic Analysis Seminar – Peter Gracar (University of Leeds)

For Bernoulli supercritical percolation on the $d$-dimensional lattice it is well understood that the infinite component exists “everywhere”. In fact, it can be shown that this component contains as a subset a Lipschitz connected hyper-surface that can be built along any of the $d-1$ possible canonical hyperplanes of $\mathbb{Z}^d$. In this talk, we will explain how one can construct a set satisfying similar properties on the Sierpiński gasket and then show how a multi-scale construction can be used to get its existence even for particle dependent percolation.

More precisely, we will consider the fractal Sierpiński gasket or carpet graph in dimension $d\geq 2$, denoted by $G$. At time $0$, we place a Poisson point process of particles onto the graph and let them perform independent simple random walks, which in this setting exhibit sub-diffusive behaviour. We will generalise the concept of particle process dependent Lipschitz percolation to the (coarse graining of the) space-time graph $G\times \mathbb{R}$, where the opened/closed state of space-time cells is measurable with respect to the particle process inside the cell. We will discuss an application of this generalised framework through the following: if particles can spread an infection when they share a site of $G$, and if they recover independently at some rate $\gamma>0$, then if $\gamma$ is sufficiently small, the infection started with a single infected particle survives indefinitely with positive probability.