Research Remco van der Hofstad
Percolation
Percolation is one of the
paradigm models in statistical physics,
displaying extremely rich critical behavior. It is a model
of a porous medium, where the materials consist of
substance and holes. My main focus
of research in the past period has been on the study of percolation
models close to criticality for highdimensional systems.
My research has primarily focussed on
 the investigation of large highdimensional
critical percolation clusters, by proving that the scaling limit of large highdimensional
critical percolation clusters is a measurevalued diffusion called
superBrownian motion and that critical percolation in sufficiently highdimensions
has an incipient infinite cluster;
 the characterization of the phase transition of percolation on large
highdimensional tori, and to prove that this phase transition is
close to the one on the ErdosRenyi random graph.
Random Graphs and Complex Networks
In the past decade, it has become clear that many real networks
share fascinating features in being small worlds and scalefree.
Such networks are typically modeled using {\it random graphs}.
Random graphs are closely related to percolation, the difference
being that random graphs tend to have finite size,
while percolation systems tend to be infinite.
The empirical findings on real networks have ignited
research on various models for complex networks.
The focus of the research of the group was the study
of distances in models of complex networks where
powerlaw degrees are observed.
My research has primarily focussed on studying distances in
random graphs. These models include

the configuration model,

various versions of generalized random graphs,

and preferential attachment models.
The goal is to show that there is different scaling in the
distances when the exponent of the power laws in the random graphs
changes. When this exponent is such that the degrees have
finite variance , then the distances grow logarithmically
with the size of the graph. When this exponent is such that the degrees have
finite mean but infinite variance , then the distances grow doubly logarithmically with the size of the graph. When this exponent is such
that the degrees have
infinite mean , then the distances remain bounded
when the size of the graph increases.
Other aspects that draw my attention is the size of the connected components
and the related phase transitions.
A key question in random graph theory related to universality,
that is, to which extent do models with similar properties show similar
behavior.
Selfinteracting Random Processes
My research on Selfinteracting random processes
has been focussed on several models:
 Onedimensional and high dimensional polymer models;
 Various selfinteracting random walks, such as reinforced random walks,
excited random walks and looperased random walks;
 Large deviations for random walk local times, and various
consequences in related models, such as the Parabolic Anderson model and
random walk in random scenery.
In this research, we make use of two key methods:

Large deviations;

Combinatorial expansion techniques for highdimensional systems, using
the lace expansion.
In one dimensions, the results focus on law of large numbers, central
limit theorem and large deviation principles for the endtoend distance of
the polymer in the limit as its length gets large. In high dimensions,
we have proved diffusive behavior of various selfavoiding walk models,
as well as of networks of such selfavoiding walks. A key question is
whether one can extend the combinatorial expansion techniques to deal with
selfinteracting stochastic processes. In the parabolic Anderson model,
the result focus on the universality properties of the solution to the
parabolic Anderson equation, when the field is i.i.d., as a function of
the tail behavior of the field.
Applications of Probability
I have always been interested in applications of probability,
particularly in electrical engineering, computer science and
theoretical physics.

With several researchers in electrical engineering, I have
contributed to the analytical study of several
multiuser detection systems , particularly
using parallel interference in DSCDMA systems
(with Marten Klok, Gerard Hooghiemstra, Anne Fey, Franck vermet and
Matthias Lowe);

We have further studied the multicarrier interference properties of
OFDM systems (with Tim Schenk, Erik Fledderus and Peter Smulders);

We are currently investigating the properties of digitaltoanalog (DAC)
converters, using a reformulation in terms of Brownian bridges.

A further topic of current research consists of the security aspects of several random
algorithms in a network. What systems performs best, and how is it related
to the topology of the network?