I am a professor at the
Bernoulli Institute for Mathematics, Computer Science and Artificial Inteligence of Groningen University. Before joining Groningen University, I held positions at Utrecht University,
at Centrum Wiskunde & Informatica (CWI), in Tel Aviv and in Eindhoven, and before that I did my doctorate in Oxford under the supervision of Colin McDiarmid. Prior to starting my doctorate, I worked for a couple of years as a statistical consultant at CANdiensten b.v.
I work in combinatorics and probability. My research interests include
random graphs, percolation, discrete and stochastic geometry, random walks, random matrices, deterministic graphs in various flavours (extremal, coloured, fractional, topological), asymptotic and probabilistic combinatorics and combinatorial game theory.
The following is a list of my journal articles, with links to preprint versions in pdf format.
Some of my research projects lend themselves well to computer experiments. Here are some visualizations I did for various talks and papers. All the pictures below have been generated (programmed) by me using R.
Confetti percolationThe confetti percolation process, sometimes also called the coloured dead leaves process, is defined as follows.
Disks of radius one arrive onto the plane according to a constant intensity space-time Poisson process. The disks are coloured either black, with probability \(p\), or white, with probability \(1-p\).
Each point of the plane assumes the colour of the first disk that lands on it.
You can think of this as the situation where confetti disks are "raining" onto a glass plate and you are looking up from underneath the glass at the pattern you see.
Alternatively, you can think of disks having rained onto the ground for a very long time. Then the "confetti rain" suddenly stops and we examine the pattern that we see on the ground.
Here are simulations corresponding to \(p=\frac14,\frac12,\frac34\), where the
pattern inside a \(200\times200\) square is shown.
In 1998, Benjamini and Schramm wanted to know for which
values of \(p\) there will exist an unbounded curve all of whose points are black.
In particular they asked whether this would be the case if and only if \(p>1/2\).
In a recent paper (number 35 in my list of publications above), I proved that this is indeed the case.
In the most general formulation, a random geometric graph is what you get if you take
a random set of points in some metric space and you join pairs of points depending on some rule (which may include additional randomness).
A special case is the Gilbert model where we take \(n\) points uniformly at random in the square (or \(d\)-dimensional hypercube) and connect two points if the distance is less than \(r\).
The model is named after E.N. Gilbert who defined a very similar model in 1961.
Here is a simulation with \(n=500\) points and \(r=.03, .06, .09\).
In my previous work I have amongst other things considered the chromatic number of and Hamilton cycles in random geometric graphs.
Voronoi percolation on the hyperbolic planeThe Voronoi percolation model is defined as follows.
We start with a Poisson process \({\cal P}\) of constant intensity on the plane.
To each point \(x\in{\cal P}\) we assign a Voronoi cell \(C(x)\), the set of all points that are closer to \(x\) than to any other point of \(\cal P\).
Finally, for each Voronoi cell we flip a coin to colour it either black (blue in the picture below), with probability \(p\), or white, with probability \(1-p\).
A pertinent question again is for which values of \(p\) there will be unbounded black curves. Or, in other words, when is there an infinite, non-repeating sequence of black Voronoi cells, each touching the next one in the sequence?
In 2006 Bollobás and Riordan showed that this is so if and only if \(p>1/2\).
Earlier, Benjamini and Schramm had already considered the situation where instead of on the ordinary, Euclidean plane, we define the Voronoi percolation model on the hyperbolic plane.
Their results highlight a number of striking differences with the Euclidean case, and they also pose a number of questions that are open to this day.
A recent paper with my PhD student Hansen (paper no. 45 in the list above)
solves one of their conjectures.
Here is a computer simulation of the hyperbolic Voronoi model
with \(p=1/2\) and intensity \(\lambda=1\), shown in the
Poincaré disk representation of the hyperbolic plane.
These simulations also featured in the poster
and a t-shirt we printed for the recent Lorentz Center workshop New Frontiers in Random Geometric Graphs.
(If you happen to wear XXL then you are welcome to collect a free t-shirt in my office.)
Here is another simulation of hyperbolic Voronoi percolation, now shown in the
Poincaré halfplane representation of the hyperbolic plabe.
In the (original, infinite) Gilbert model, we take a Poisson process \({\cal P}\) on the plane, and we create a graph by joining any two points by an edge that have distance less than \(r\). This of course also makes sense in the hyperbolic plane. Here is a computer simulation, with \(r=1\) and intensity \(\lambda=5\) of the Poisson process, shown on the left in Poincaré disk representation and on the right in the native representation.
A different way of generating a random graph in hyperbolic space was introduced
recently by Krioukov-Papadopoulos-Kitsak-Vahdat-Boguñá.
Perhaps rather surprisingly, this random graph model has features that are
usually attributed to complex networks, including a power law degree distribution and clustering.
Besides the number of nodes, the model has two parameters, \(\alpha\) and \(\nu\). Roughly speaking \(\alpha\) controls the exponent of the powerlaw and \(\nu\) controls the average degree.
For a more detailed definition of the model see my papers with Bode and Fountoulakis above.
Here are computer simulations corresponding to 5000 nodes with \(\alpha=.9,\nu=.2\) (left) and \(\alpha=.51,\nu=.2\) (right), shown in the native model of the hyperbolic plane.
Together with Michel Bode, Nikolaos Fountoulakis, Dieter Mitsche, Merlijn Staps, Markus Schepers and Pim van der Hoorn (in various constellations) I have written several papers on this model.
In Groningen, there is an active Probability Seminar.
Formerly, I was co-organizer of the Stochastics Seminar and the
Mathematics colloquium in Utrecht.
I am / have been (co-) organizer of the following conferences and workshops:
This academic year, I am teaching the minor course course percolation theory, the second year course discrete mathematics, the first year course introduction to graph theory and the master course Combinatorial Mathematics A.
E-mail: | tobias dot muller at rug dot nl |
Address for correspondence: | Rijksuniversiteit Groningen Bernoulli Institute P.O.box 407 9700 AK Groningen The Netherlands. |