Conference Agenda

We study survival and extinction of a long-range infection process on a diluted one- dimensional lattice in discrete time. The infection can spread to distant vertices according to a Pareto distribution, however spreading is also prohibited at random times. We prove a phase transition in the recovery parameter via block arguments. This contributes to a line of research on directed percolation with long-range correlations in nonstabilizing random environments.

Recently, there has been increasing interest in interacting particle systems on evolving random graphs, respectively in time evolving random environments. In this talk we present results on the contact process in an evolving edge random environment on infinite (random) graphs. We in particular consider (infinite) Galton-Watson trees as the underlying random graph. We focus on an edge random environment that is given by a dynamical percolation whose opening and closing rates and probabilities are degree dependent. Our results concern the dependence of the critical infection rate for weak and strong survival on the random environment.

We suggest a non-asymptotic matrix perturbation-theoretic approach to get sharp bounds on the expected meeting time of random walks on large (possibly random) graphs. We provide a formula for the expected meeting time in terms of the singular value decomposition of the diagonally killed generator of a pair of independent random walks, which we view as a perturbation of the generator. Employing a rank-one approximation of the diagonally killed generator as the proof of concept, we work out sharp bounds on the expected meeting time of simple random walks on sufficiently dense Erdős-Rényi random graphs.

In the broadcasting problem on trees, a $\{-1,1\}$-message originating in an unknown node is passed along the tree with a certain error probability $q$. The goal is to estimate the original message without knowing the order in which the nodes were informed. We show a connection to random walks with memory effects and use this to develop a novel approach to analyse the majority estimator on random recursive trees. With this powerful approach, we study the entire group of very simple increasing trees as well as shape exchangeable trees together. This also extends Addario-Berry et al. (2022) who investigated this estimator for uniform and linear preferential attachment random recursive trees.