Plenary Lectures

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 first give an overview over recent results. Then, we in particular consider (infinite) Bienaymé-Galton-Watson (BGW) trees as the underlying random graph. Here, 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 impact of these parameters on the critical infection rate for weak (global) and strong (local) survival of the infection.

Specifically, we establish conditions under which the contact process undergoes a phase transition: For a general connected locally finite graph we provide sufficient conditions for the critical infection rate to be strictly positive. Furthermore, in the setting of BGW trees, we provide conditions on the offspring distribution as well as on the speed and connection probabilities that the process survives strongly with positive probability for all positive values of the infection rate. In particular, if the offspring distribution follows a power law (or has a stretched exponential tail) and the connection probability is given by a product kernel (or a maximum kernel) and the update speed exhibits polynomial behaviour, we provide quite a complete characterisation of the contact process' behavior.

This talk is based on joint work with Natalia Cardona-Tobon (Universidad Nacional de Colombia, Bogotá), Marcel Ortgiese (University of Bath) and Marco Seiler (University of Frankfurt).

This work was motivated by natural questions related to oriented percolation on a layered environment that introduces long range dependence. As a convenient tool, we are led to deal with questions on the strict decrease of the percolation parameter in the oriented setup when an extra dimension is added.

Joint work with Daniel Ungaretti and Maria Eulália Vares.

We discuss a class of nonlinear exchange dynamics for interacting spin systems, inspired by Boltzmann-type equations from kinetic theory. These dynamics provide a nonlinear Monte Carlo approach for sampling Gibbs measures under prescribed magnetization or density constraints. We focus on the Ising and hard-core models and present convergence results showing exponential relaxation to equilibrium in the weak-interaction and low-density regimes, respectively, with optimal decay rates independent of the initial condition. The proofs are based on novel coupling techniques as well as ideas from the Kac program in kinetic theory, including uniform functional inequalities for the associated particle systems. We also discuss some possible algorithmic applications.

The counting of partitions of an integer n under different restrictions appears in many areas, and has been considered from several viewpoints. A family of random variables can be naturally associated to the generating function of the corresponding partition function, and it turns out that those are sums of independent random variables, which are asymptotically normal when conveniently normalized. The asymptotic behaviour of the number of partitions is then related to local limit theorems. We investigate in particular the asymptotic behaviour of the number of partitions of n with k colors, both in the case of unrestricted partitions, and in that into distinct parts. In the case that k grows linearly with n, we show that an asymptotic expansion follows from the classical expansions of the local limit theorems by Gnedenko and Kolmogorov. In the case that k is fixed, precise asymptotic results are deduced by techniques involving expansions in cumulants.

Based in joint work with A. Meyroneinc and Y. Vargas.