Some further details on individuals

Pre-amble:
There is a healthy group of substantial size working on probability in Cambridge, including the staff members listed below, together with `parts' of Frank Kelly, Richard Weber, and Doug Kennedy. At any given time, we have typically several PhD students and postdoctoral fellows funded by the EPSRC, EU, colleges, and other sources. Present and recent postdocs include Alan Stacey, James Martin, Inez Armendariz, Oliver Johnson, Marcelo Piza, Michail Loulakis. Each of these is an independent researcher or is the process of developing into one, with publications not listed here (but listed on MathSciNet).

Yuri Suhov has work which will described by Frank Kelly under the heading of the Mathematics for Operations Research.

The main lab web page is to be found here.

Geoffrey Grimmett:
1. The main thrust of my current work is to construct a systematic theory of percolation/Ising/Potts models using the random-cluster model as a unifying tool. I have several papers on this theme including

The stochastic random-cluster process and the uniqueness of random-cluster measures, Annals of Probability 23 (1995) 1461-1510

The random-cluster model in Probability on Discrete Structures, ed. H. Kesten, Encyclopedia of Mathematical Sciences, vol. 110, Springer, 2003, 73-123

Rigidity of the interface in percolation and random-cluster models with Guy Gielis Journal of Statistical Physics 109 (2002) 1-37

and am currently writing a book (I am up to ~250 pages with 100 to go). I see this work as continuing my work on percolation and related models, which reached some sort of conclusion with the publication of

Percolation, Second edition, Springer 1999.

2. There is a beautiful problem of Lorentz (1905) on understanding the behaviour of a certain dynamical process in a random environment, namely a model for the movement of a light paricle through a field of heavy particles. There is a stochastic relaxation of this problem which has led to some questions of beauty. See:

Stochastic pin-ball in Random Walks and Discrete Potential Theory, ed. M. Picardello and W. Woess, Cambridge University Press, 1999, 205-213

for a survey.

3. Then there are several other recent things on random triangles and Poisson approximation, quantum random walk, oriented percolation and random walk, etc.

For further details and all papers.

Ostap Hryniv:
1. The main direction of my research includes mathematical theory of phase transitions, especially statistics of interfaces. One of the recent results here is the paper

Self-avoiding polygons: Sharp asymptotics of canonical partition functions under the fixed area constraint (with D.Ioffe); mp_arc 01-434, to appear in MPRF,

which gives a complete description of droplets bounded by self-avoiding polygons; the obtained local limit theorem is valid in the whole subcritical region.
One of my current projects is to generalize this result to the case of the Ising model in two dimensions.

2. The paper

Surface tension and the Ornstein-Zernike behaviour for the 2D Blume-Capel model (with R.Kotecký) J. Stat. Phys. 106 (3-4): 431-476, 2002

develops an approach to rigorous probabilistic treatment of statistical properties of interfaces in systems whose spins take more than two values.
Its continuation aims at extension of the result to general low-temperature models in the framework of Pirogov-Sinai theory.

3. According to physicists' beliefs, many critical phenomena exhibit certain universality. In the paper

Universality of Critical Behaviour in a Class of Recurrent Random Walks. (with Y.Velenik); math.PR/0310217, submitted

we discuss this issue for a toy model of critical prewetting.

For further details.

James Norris:
1. The main project I have been working on is a derivation of Smoluchowski's coagulation equation from a system of (Stokes-Einstein-)Brownian particles subject to coagulation on collision. This work is essentially complete but might count as a future plan. There are many (phyically motivated) variations on the problem, for example the case of Ornstein-Uhlenbeck particles, which appear susceptible to the same general strategy. Ines Armendariz is working on this. We are also working in collaboration with Chemical Engineers who are developing stochastic numerical schemes for coagulation processes. Earlier work on mean-field type models was published in

Cluster coagulation. Comm. Math. Phys. 209 (2000), no. 2, 407--435.

2. A second project, in collaboration with Richard Darling, concerns the structure of large random combinatorial structures. We are able to work with a rather general hypergraph model, which appears to offer scope for genuine mathematical modelling. The key idea here is the use of stochastic process methods such as fluid limits and Gaussian approximations to pass to a simplifying limit before doing any calculations. This is the context of Christina Goldschmidt's PhD, which established in particular some new scaling properties of near-critical random hypergraphs. The basic paper is math.PR/0109020.

3. Going back a few years, the paper

Heat kernel asymptotics and the distance function in Lipschitz Riemannian manifolds. Acta Math. 179 (1997), no. 1, 79--103.

answered a long-standing open problem about the short-time behaviour of heat flow. Using similar methods, the paper

Long-time behaviour of heat flow: global estimates and exact asymptotics. Arch. Rational Mech. Anal. 140 (1997), no. 2, 161--195.

made novel connections between homogenization methods and heat kernel estimates, and gave new variational characterizations of effective constants.

4. A little bit further back,

Twisted sheets. J. Funct. Anal. 132 (1995), no. 2, 273--334

is a geometric theory of two-parameter hyperbolic SPDE. This is a powerful tool for the path-space analysis of Malliavin and others.

For further details.

Chris Rogers: In recent years, Professor Rogers and his coworkers have contributed a number of important papers to the development of mathematical finance; among the more significant of these are:

`Monte Carlo valuation of American options.' Mathematical Finance 12, 2002, 271--286

`Duality in constrained optimal investment and consumption problems: a synthesis'. CIRANO Workshop on Mathematical Finance and Econometrics, June 2001.

`Robust hedging of barrier options' (with H. M. Brown and D. G. Hobson). Mathematical Finance 11, 2001, 285--314.

`Markov chains and the potential approach to modelling interest rates and exchange rates' (with F. A. Yousaf). Mathematical Finance - Bachelier Congress 2000, ed. H. Geman, D. Madan, S. R. Pliska, & T. Vorst, Springer 2002.

The first of these presents a completely novel approach to the pricing of American options, allowing upper bounds on prices to be established; this has had considerable impact on industry practice as well as academic research. The second is a unification and simplification of a substantial recent literature on constrained optimal investment and consumption problems of various types. The third can be seen as a first attempt to understand the use of `quasi-static' hedging techniques, and raises a number of intriguing questions. The final paper develops an approach pioneered by Professor Rogers in the mid-90s to modelling in fixed income and FX markets; it shows that the approach is viable in practice, and subsequent unpublished developments are even more promising.

For further details and all papers.

Yuri Suhov:
1. M. Kontsevich and Y. Suhov. Statistics of Klein polyhedra and multi-dimensional continued fractions (with M. Kontsevich). In: {\it Pseudoperiodic Topology} (eds V. Arnol'd, M. Kontsevich and A. Zorich). Amer. Math. Transl. Ser. 2, vol. {\bf 197}. Providence, R.I.: American Mathematical Society, 1999, pp. 9--28

The standard continuous fraction representation $$x=\frac{1}{a_0+\frac{1}{a_1+\frac{1}{a_2+\ldots}}}$$ of a number $x\in (0,1)$ generates a dynamical system (a shift $(a_0,a_1,a_2,\ldots )\mapsto (a_1,a_2,\ldots )$) with an invariant probability measure known as a Gauss, or Gauss-Kuzmin measure (unrelated to the famous Gaussian distribution). Statistical properties of the sequence $(a_0,a_1,a_2,\ldots )$ relative to the Gauss-Kuzmin distrubution are well-known and play an important role in several applications, including rational approximations of a given real number. Multi-dimensional generalisations of continued fractions lead to reach and intereseting geometric structures and are important, in particular, in rational approximations of a given real vector. The question of studying statistical properties of multi-dimensional continued fractions was posed, by, among others, V. Arnol'd. He stressed importance of studying `typicality' of geometric patterns relative to a `natural' measure replacing the Gauss-Kuzmin measure in multi-dimenmsional situation. This patterns arise when one deals with a random simplicial cone and a lattice inside this cone, with respect to the above measure.

2. As many of us, I prefer to talk about new things. In a paper with Chulaevski (in progress) we extend the Anderson localisation theory to the case of several particles, which is the first rigorous result in this direction in the world literature.

For further details.

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