Asymptotic Analysis of Random Matrices and Exactly Soluble Models

Some magnets can be imagined as a collection of small elementary magnets ('compas needles' called spins). If an outside magnetic field is applied they tend to aline themselves in the direction of the field. If the magnetic field is removed,they tend to become disoriented again. One of the important questions is to know how quickly the alignment is destroyed. It depends on the type of a magnet and the temperature.This information is encoded in the so called correlation functions of the spins.Correlation functions tell us how well the spins 'feel' one another.If we pick one spin in the magnet,one can ask the probability that another spin points in the same direction. This probability is one of the examples of the correlation functions. It is in general impossible to calculate correlation functions exactly. However, in some simple magnets, it is possible to calculate them asymptotically. This last wordmeans, in the example above, that we may be able to calculate the correlation function of two spins when the distance between them becomes large.Such asymptotic calculations are the subject of this proposal.