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Interaction and Growth in Complex Stochastic Systems

21st July 2003 to 19th December 2003

Organisers: Professor E Bolthausen (Zurich), Professor K Khanin (Heriot-Watt), Professor G Lawler (Duke), Dr JR Norris (Cambridge), Professor YM Suhov (Cambridge)

Programme Theme

Random processes are often understood by means of an asymptotic in which the number of basic random components becomes large. Classical examples of behaviour which can be manifest in such a limit are the law of large numbers and the central limit theorem. These two examples apply in contexts where the interaction between typical random components becomes negligible in the limit. However, there are many extremely important physical models, where such an asymptotic decorrelation does not hold. Such models correspond to a critical behaviour where interaction is strong and statistical scaling properties are highly nontrivial. Understanding of asymptotic properties for such systems is one of the most important problems in the modern probability theory. Renormalization group ideology which was developed by physicists in the last 30 years predicts appearance of the universal scaling and conformal invariant random fields characterizing statistical behaviour in critical situations. However, rigorous results in this direction were out of reach until very recently.

The programme will focus on problems from a number of contexts where strong random interactions give rise to non-classical limiting behaviour. These will include random walk in a random environment and interactive random walk, interfaces in growth models and equilibrium statistical mechanics, coagulation and fragmentation, conformally invariant scaling limits, random matrices.

Powerful new techniques are beginning to make possible the computation of exact asymptotics along with a rigorous understanding of convergence. The range of possible limiting behaviour is rich and introduces to probability new connections with other areas of mathematics and physics. The programme will bring together many aspects of this work, including its applications, from statistical mechanics to applied probability.

Final Scientific Report: 

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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons