# 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