Statistical Mechanics and Field Theory

点击翻译按钮获取中文摘要

1. The purpose of this project is to enhance our understanding of fundamental problems instatistical mechanics and probability theory through the use of mathematical methodsinvolving supersymmetry and the renormalization group. Of particular interest arestatistical models with a geometrical flavor, such as branched polymers and self-avoidingwalks. Basic questions for these models involve, for example, determining the rate atwhich the size of a sample grows with the number of monomers or steps. Such divergencesare governed by critical exponents. The values of the exponents may be determined in oneof two ways, either through dimensional reduction (which relates exponents to knownexponents in related systems in fewer dimensions), or through the renormalization group(which calculates the flow of couplings with the length scale as the microscopicstructures of the problem are averaged out). Supersymmetry plays an important role inboth methods.Specific goals are: (a) to continue to develop a mathematical theory of dimensionalreduction and apply it to new models, such as directed branched polymers; and (b) toconstruct an exact renormalization group flows for such systems.2. Research activities in this proposal will involve collaborations and exchanges ofideas with mathematicians and physicists in a wide range of fields, including probability,analysis, statistical mechanics, and quantum field theory. Graduate student support, andgraduate student participation in scientific meetings, will integrate research andeducation and help train a new generation of mathematicians in the methods of fieldtheory, supersymmetry, and the renormalization group. These activities will in the longrun lead to enhanced interaction between mathematics and physics and to new ideas in bothfields.