# Research Interests: Mathew Penrose

My research activities have covered a variety of topics in modern probability theory, often motivated by questions from the physical sciences, under the general headings of stochastic analysis, extreme value theory, percolation, interacting particle systems and geometric probability,

This page contains a (hopefully!) non-technical description of these activities. Some of them are recent, others less so. The more recent activities are described first. While the emphasis here is on completed work, there is inevitably an overlap between topics described here, and possible future 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.

## Geometric probability

My work in papers such as [R19], [R22], and [R27], has been concerned with connectivity properties of graphs obtained by placing points at random in the plane, say, and connecting any two points separated by a distance of at most r. I am also interested in other properties of these `geometric random graphs', for example the vertex degrees [O4]; these graphs can be used for example in tests for uniformity of data, or as models for electronic or social networks. Further research on these graphs [O5] has been motivated by issues of computational complexity.

My work in papaers such as [R18] and [R19] has concerned the structure of the Euclidean minimal spanning tree. This is the graph connecting a given set of points in d-dimensional space with minimal total length, and is of interest in computer science, the physical sciences and in statistics. The structure of the minimal spanning tree on random points is of interest, for example because of certain statistical tests (see [R22], [R25]), and is only partially understood. The random points could be uniformly distributed on the unit cube, or have some general probability density function.

In [R21], I worked on the minimal spanning tree in the n-cube, a discrete structure with some analogies to high-dimensional Euclidean space. I hope to continue to work on such discrete structures in the near future. There may be applications in mathematical genetics.

## Percolation and interacting particle systems

Percolation theory 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 this area 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 the topic, and [R16] for some results on high-dimensional asymptotics.

For an account of aspects of the related topic 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. Often, particle system models, and also percolation, exhibit a phase transition at a critical value of some parameter, such as the rate of infection.

Another lattice-based mathematical model is the self-avoiding walk. Imagine a particle moving around in a random way among neighbouring sites in a 3-dimensional lattice, constrained never to visit the same site twice. This has been studied as a model for long polymer chains. An important quantity in self-avoiding walk is the so-called `connective constant'.

I have also worked on a variety of lattice-based models of the type described above. Usually the precise critical value or connective constant is not analytically known. In [R8], [R11], [R15], and [R17], I developed exact asymptotics for these critical values in certain limiting cases for the lattices.

## Extreme value theory

The research in [R1], [R4], and [R6] concerns the study of the evolving maximum of a collection of randomly evolving quantities, for example share prices. The maximum of is a single evolving quantity, which could represent the value of a financial instrument giving someone a choice between the shares. Various aspects of the behaviour of this evolving maximum is studied in these papers, and also in my Ph.D. Thesis [O1]. In addition, more recent work in [R19] [R22] [R27], [R24] could be described as `geometric extreme value theory.'

## Brownian motion and stochastic analysis

The research in [R10, R12] is concerned with a mathematical model for diffusion-controlled chemical reaction, with molecules performing Brownian motion and annihilating on contact. A description of these reactions for ``sparse'' reagents is described, using a probabilistic tool known as the theory of U-statistics.

The main topic of my PhD thesis [O1], also the subject of in [R2,R3], is the behaviour of mathematical Brownian motion. Of special case in this work was the study of self-avoiding Brownian paths in three-dimensional space, a continuous-space analogue of the self-avoiding walk described above. In [R2] is a result about the non-existence, in some sense, of these self-avoiding paths.

One major application of the theory of Brownian motion and stochastic analysis is in the study of certain partial differential equations. In [R9], the theory of brownian motion is used to analyse some equations arising in statistical physics, and to obtain new information on the deviation from the ideal gas law for a particular model of molecular interaction. Some probabilistic questions concerned, with the three-dimensional Brownian bridge (Brownian motion constrained to be at a specific position at a specified time) were studied in [R7].

## My favourite theorem

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

Last update 1 June 1999