Day 1 (April 6) @ 11:00–12:30

Henk Don (Radboud Universiteit)

A lower bound for connection probabilities in critical percolation

In this talk we consider critical percolation on $\mathbb{Z}^d$. In particular we are interested in the probability to have an open path from the origin to some point $x$ as a function of $x$. In dimensions 3-6 not much is known about this function, as the techniques for dimension two and those for high dimensions fail. We will present a lower bound on the connection probabilities in these intermediate dimensions. (Joint work with Rob van den Berg)

Luca Avena (University of Leiden)

Explorations of networks through random rooted forests

In the 1990s David Wilson introduced a simple algorithm based on loop-erased random walks to sample uniform spanning trees and, more generally, rooted weighted trees and forests spanning a given graph. I will consider the probability measure obtained when Wilson’s algorithm is used to sample rooted forestswith a scale-parameter controlling the number of trees.The resulting forest measure has a rich, flexible and explicit mathematical structure which makes it a powerful tool to design algorithms to explore networks in a dynamic fashion by tuning the scale-parameter. 

I this lecture I will focus on fundamental aspects of this measure and its relations with other objects of relevance in probability and statistical physics. In particular, I plan to describe the main static and dynamic properties of associated observables (e.g. set of roots, induced partition) and discuss some progress in understanding related asymptotics.

At the end I will briefly mention how this forest can be used to design algorithms to analyze network-based data sets.

Daniel Valesin (Rijksuniversiteit Groningen)

Percolation on stationary distributions of interacting particle systems

We discuss certain models of site percolation on the d-dimensional integer lattice in which long-range dependence among the states of vertices is present. In the presence of such dependence, proving that a percolation phase transition occurs can be challenging.  In this talk we will address models in which the states are sampled from probability distributions obtained from the equilibrium states of interacting particle systems. We will discuss two important classes of interacting particle systems, namely the voter model and the contact process. In both cases, we prove that (under certain assumptions on the system dynamics) a percolation phase transition is present. Joint work with Balázs Ráth (BME, Hungary).