Stochastic Activity Month

February 8

2012

Lecture Afternoon

""Applications for Scaling Limits"

SPEAKERS

Ronald Meester (VU Amsterdam)

Florian Simatos (CWI-Eurandom)

Pieter Trapman (Stockholm Uinversity)

REGISTRATION

For organizational purposes, please register online: Registration form

PROGRAMME

ABSTRACTS

Florian Simatos

On the scaling limits of regenerative processes

In this talk I will first present a general result that relates the convergence of regenerative processes to the convergence of their excursions. I will then present two non-trivial applications of this method that arise in queueing theory.
(this talk is based on on-going joint works with Sem Borst, Amaury Lambert and Bert Zwart)

Pieter Trapman (University of Stockholm)

The growth of the infinite long-range percolation cluster and an application to spatial epidemics

Consider long-range percolation on $\mathbb{Z}^d$, where the probability that two vertices at distance $r$ are connected by an edge is given by $p(r) = 1-\exp[-\lambda(r)] \in (0,1)$ and the presence or absence of different edges are independent.
Here $\lambda(r)$ is a strictly positive, non-increasing regularly varying function. I will discuss the growth of the number of vertices that are within graph-distance $k$ of the origin, $|\mathcal{B}_k|$, as $k \to \infty$.
Conditioned on the origin being in the (unique) infinite cluster, non-empty classes of non-increasing regularly varying functions are identified, for which respectively
- $|\mathcal{B}_k|^{1/k} \to \infty$ almost surely,
- there exist $1 <a_1 < a_2 < \infty$ such that $\lim_{k\to \infty} \mathbb{P}(a_1<|\mathcal{B}_k|^{1/k}< a_2) = 1$,
- $|\mathcal{B}_k|^{1/k} \to 1$ almost surely.

This result can be applied to spatial epidemics. In particular, regimes are identified for which the basic reproduction number, $R_0$, which is an important quantity for epidemics in unstructured populations, may have a useful counterpart in spatial epidemics.

this talk is based on:
- P. Trapman (2010), The growth of the infinite long-range percolation cluster, Annals of Probability.
- S. Davis, P. Trapman, H. Leirs, M. Begon and J.A.P. Heesterbeek (2008), The abundance threshold for plague as a critical percolation phenomenon, Nature.

PRESENTATION

Ronald Meester ( VU Amsterdam)

Scaling limits and continuous curves in fractal percolation

Scaling limits are distributional limits, whereas fractal percolation is a process which creates a random fractal as a pointwise limit. In this lecture I will explain how ideas in scaling limits can be used to obtain information about the (pointwise) limiting set in fractal percolation, especially about the existence of Holder continuous curves. Although the subject of scaling limits in general tends to be rather technical, the lecture will be very non-technical.