Statistical Mechanics and Field Theory

University Of Virginia Main Campus, Charlottesville VA

Investigators

Abstract

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