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 high-dimensional systems.

My research has primarily focussed on
  1. the investigation of large high-dimensional critical percolation clusters, by proving that the scaling limit of large high-dimensional critical percolation clusters is a measure-valued diffusion called super-Brownian motion and that critical percolation in sufficiently high-dimensions has an incipient infinite cluster;
  2. the characterization of the phase transition of percolation on large high-dimensional tori, and to prove that this phase transition is close to the one on the Erdos-Renyi 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 scale-free. 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 power-law degrees are observed.

My research has primarily focussed on studying distances in random graphs. These models include
  1. the configuration model,
  2. various versions of generalized random graphs,
  3. 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.

 

Self-interacting Random Processes

My research on Self-interacting random processes has been focussed on several models:
  1. One-dimensional and high dimensional polymer models;
  2. Various self-interacting random walks, such as reinforced random walks, excited random walks and loop-erased random walks;
  3. 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:
  1. Large deviations;
  2. Combinatorial expansion techniques for high-dimensional 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 end-to-end distance of the polymer in the limit as its length gets large. In high dimensions, we have proved diffusive behavior of various self-avoiding walk models, as well as of networks of such self-avoiding walks. A key question is whether one can extend the combinatorial expansion techniques to deal with self-interacting 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.
  1. With several researchers in electrical engineering, I have contributed to the analytical study of several multiuser detection systems , particularly using parallel interference in DS-CDMA systems (with Marten Klok, Gerard Hooghiemstra, Anne Fey, Franck vermet and Matthias Lowe);
  2. We have further studied the multicarrier interference properties of OFDM systems (with Tim Schenk, Erik Fledderus and Peter Smulders);
  3. We are currently investigating the properties of digital-to-analog (DAC) converters, using a reformulation in terms of Brownian bridges.
  4. 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?