# STAR

## STAR

Henk Don

#### 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
(Leiden University)

#### 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 forests.The resulting forest measure has a rich, flexible and explicit mathematical structure which makes it a powerful tool to design algorithms to explore networks.

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 (e.g. RW-Green’s kernel, random-cluster model, determinantal processes). 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)