Statistical Mechanics and Field Theory

1.这个项目的目的是通过使用涉及超对称性和重整化群的数学方法来加深我们对统计力学和概率论基本问题的理解。特别令人感兴趣的是具有几何色彩的统计模型，例如支化聚合物和自回避漫步。这些模型的基本问题包括，例如，确定样品的大小随着单体或步骤的数量而增长的速度。这种分歧是由临界指数决定的。指数的值可以用两种方法中的一种来确定，要么通过降维(将指数与较少维度的相关系统中的已知指数联系起来)，要么通过重整化群(随着问题的微观结构的平均化，用长度尺度计算耦合的流量)。超对称性在这两种方法中都扮演着重要的角色。具体的目标是：(A)继续发展降维的数学理论，并将其应用于新的模型，如定向支化聚合物；以及(B)为这类系统构造精确的重整化群流。这项建议中的研究活动将涉及与数学家和物理学家在广泛领域的合作和交流，包括概率、分析、统计力学和量子场论。研究生支持和研究生参与科学会议，将把研究和教育结合起来，帮助培养新一代数学家，使他们掌握场论、超对称性和重整化群的方法。从长远来看，这些活动将加强数学和物理之间的互动，并在这两个领域产生新的想法。

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.