The critical behaviour of random systems

随机系统的临界行为

• 批准号: 9351-2007
• 负责人: Slade, Gordon
• 金额: $ 3.72万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2007
• 资助国家: 加拿大
• 起止时间: 2007-01-01 至 2008-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=363167
• 关键词: critical; behaviour; random; systems

Mathematical models such as the self-avoiding walk and percolation are of considerable interest in statistical mechanics, probability theory, and combinatorics.  These models have been studied for about half a century, and remain a fascinating source of problems for mathematicians.  They have also been studied by physicists and chemists, as models of polymer molecules and of phase transition.  Their significance for mathematics is currently greater than ever, and the subject is flourishing.The main mathematical problem is to prove theorems which shed light on the critical exponents and critical scaling limits which govern the large scale behaviour of these models.  An important mathematical method for such problems is the so-called lace expansion, and one thrust of this proposal is to further develop and apply the lace expansion to these and other problems.  Another important mathematical method for such problems is the renormalization group.  A second thrust of this proposal is to develop a new and flexible mathematically rigorous approach to the renormalization group, and to apply it to study 4-dimensional self-avoiding walks.

数学模型，如自避行走和渗流，在统计力学、概率论和组合学中有着相当大的意义。这些模型已经被研究了大约半个世纪，并且仍然是数学家们感兴趣的问题的来源。物理学家和化学家也对它们进行了研究，作为聚合物分子和相变的模型。它们在数学上的重要性现在比以往任何时候都要大，这门学科正处于蓬勃发展的时期，主要的数学问题是证明控制这些模型大尺度行为的临界指数和临界标度限的定理，这是一种重要的数学方法，问题的第一个关键是所谓的花边展开，这个建议的一个重点是进一步发展和应用花边展开到这些和其他问题。另一个重要的数学方法，这样的问题是重整化群。这个建议的第二个重点是发展一个新的和灵活的数学严格的方法重整化群，并将其应用于研究4维自避免行走。