MARK KAC SEMINAR

In this talk I will investigate what is called `supercritical sharpness’ for a class of spatial percolation models. Percolation models were originally invented as toy models to study phase transition: the simplest model is on the d-dimensional integer lattice $\mathbb Z^d$, where we keep each nearest neighbor edge with probability p. What we see is that there is a critical probability $p_c$ below which there is no infinite component and above which there is a unique infinite component. The locality conjecture of Schramm says that the critical probability is locally determined, i.e., for a sequence of (infinite) graphs G_n where the local neighborhoods converge in distribution to a limiting graph $G$, $p_c(G_n)$ converges to $p_c(G)$.

Locality has strong relations to `supercritical sharpness’, which is the fast decay of cluster size distribution of finite clusters all the way down to $p_c$. In this talk we will study percolation models where long edges are also allowed, with a probability that decays with the distance, and establish the cluster size decay which may be different from the decay of nearest neighbor models. Joint work with Joost Jorritsma and Dieter Mitsche, and also Yago Moreno Alonso.

We consider a Poisson point process with constant intensity in $ \mathbb{R}^d $ and independently color each cell of the resulting random Voronoi tessellation black with probability $ p $. The critical probability $ p_c(d) $ is the value for $ p $ above which there exists almost surely an unbounded black component and almost surely does not for values below. We show that $ p_c(d)=(1+o(1)) e d^{-1}2^{-d} $, as $ d\to\infty $. We also obtain the corresponding result for site percolation on the Poisson-Gabriel graph, where $ p_c(d)=(1+o(1))2^{-d} $.

From the point of view of statistical physics, it is natural to expect that solutions to PDEs should have a probabilistic interpretation. The most famous example of this is how the heat equation can be solved using random walks. In this talk, we will consider first-order conservation PDEs and show how their solutions can be represented using multi-type branching process. The connection between the two will be illustrated using random graphs. If time permits, we will also show how this interpretation can be used for numerical simulations. This talk is based on joint work with Ivan Kryven.