#
Research Interests: Mathew Penrose

Research interests cover a variety of topics
in modern probability theory, often motivated
by questions from the physical sciences, under the general
headings of geometric probability, interacting particle systems,
percolation, stochastic analysis, extreme value theory,
This page contains a (hopefully!) non-technical description of
recent activities and ongoing projects, with an emphasis
on subject with potential for collaborative work or PhD projects.
References in [square brackets] are to my
list of publications.
References such as [R18] are to my list of
refereed papers, while those such as to [O3] are to other work.
For a more detailed description of research activities up
to 1999, see
1999 research descriptiion.

##
Percolation and interacting particle systems

For an account of aspects of interacting particle
systems, see my set of lecture notes [O3].
Typically, these systems involve a collection of particles
living on a lattice, evolving in a random way, with neighbouring
sites on the lattice interacting. For example, certain
models of spatial epidemics can be formulated naturally in this way.
Of particular current interest
are systems modelling the sequential deposition of particles
onto a surface. Such systems are important in
physical chemistry and biochemistry, and a current
interest is in backing up the numerous existing simulation studies
with some rigorous mathematical theory.
Papers
[R33] and
[06] and
make a start in this direction, and there is certainly scope
for future research here.

The related topic of
percolation is a simple and popular stochastic model
for disordered physical systems exhibiting phase transitions
(i.e. sudden changes in the large-scale structure as a parameter is varied).
Much of my work in these areas has been for
continuum models, which have been growing in popularity, being
often more realistic than lattice-based models. See
[R5]
for an introduction to continuum percolation.

##
Geometric probability

Consider a set of points placed at random in the plane,
with some rule specified for connecting pairs
of points which lie close to each other, to make a graph.
For example, one could connect any two points separated by a distance
of at most *r*. Or one could connect each point to its nearest
neighbour, and there are many other ways to connect points.
Such graphs arise in mathematical modelling, for example
of communications or social networks, and statistical testing,
for example in tests for uniformity of data.
I am interested in studying the properties of such graphs.
For just one example of such graphs, see [O4];
Results on such graphs with potential applications to statistical tests
can be found for example in (see [R22], [R25]),
and [O4];
Further research on these graphs
[O5]
has been motivated by issues
of computational complexity. This work is concerned
with ``efficient orderings'' of such graphs,
and there is certainly scope for further reserch in this direction.
It is also possible to consider analogous graph constructions
in non-eulclidean spaces, for example high-dimensional
discrete spaces. This could be an area of future research,
extending work in
[R21].
There may be applications in mathematical genetics.

##
My favourite theorem

According to Joel Spencer, `every mathematician has a result he is most pleased
with.' Here's mine.
Last update 22 Feb 2001