Renormalisation group and critical phenomena

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Mathematical models of phase transition and critical phenomena are major topics in statistical mechanics and probability theory, with important implications both within and outside mathematics. Their analysis requires the understanding of an infinite number of random variables whose mutual dependence is strong enough to cause long-range correlations. The search for new mathematics to describe this scenario is at the cutting edge of probability theory, and is the topic of this proposal. Two fundamental models of critical phenomena will be studied: lattice spin models of ferromagnetism, and the self-avoiding walk model of linear polymers. A good reason for studying simplified mathematical models is that critical phenomena exhibit universality: the macroscopic critical behaviour is predicted to be independent of the fine details of how the model is defined microscopically. Universality allows physically meaningful conclusions to be drawn from models which at first sight may seem overly simplified. An important aspect of the universal critical behaviour is encompassed by the critical exponents that govern quantities such as correlation length and susceptibility near the critical point. Both the spin and the self-avoiding walk models can be defined on a lattice of any dimension d. The applicant and his collaborators have recently developed a mathematically rigorous and general version of the renormalisation group (RG) method, to analyse dimension d=4. The primary thrust of this proposal is to extend this RG method to analyse critical phenomena below dimension 4. To understand lower dimensions, fractional dimensions have been used in the physics literature for the computation of critical exponents, since the pioneering work of Wilson and Fisher on the epsilon expansion in the early 1970s. The use of fractional dimensions is mathematically problematic, but an alternate approach is to mimic the use of fractional dimensions by considering long-range models based on a fractional power of the Laplace operator. The primary thrust of this proposal is to extend the RG method to obtain a mathematically rigorous version of the epsilon expansion in dimensions d=1,2,3, for such long-range models. This will provide the first results of this kind on a Euclidean lattice. The methods will then be used to prove existence of and compute the values of other critical exponents, for the correlation length and for various critical correlation functions including the two-point function, and to study critical scaling limits and universality. A long-term goal is to apply the methods to analyse the end-to-end distance for the self-avoiding walk. This research will provide a rigorous mathematical foundation for important problems of physical interest, whose mathematical understanding has remained unresolved for more than 40 years.