The critical behaviour of random systems

数学模型，如自避行走和渗流是相当大的兴趣，在统计力学，概率论和组合。 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的一个迷人的问题来源。 它们也被物理学家和化学家研究，作为聚合物分子和相变的模型。 它们对数学的重要性目前比以往任何时候都要大，这一学科正在蓬勃发展。 主要的数学问题是证明定理，揭示了关键指数和关键的缩放限制，这些模型的大规模行为。 这个建议的主旨是进一步发展一个新的和灵活的数学上严格的重正化组的方法，并应用它来获得详细的信息4维自避免行走。 该方案的另一个目的是进一步发展和应用花边展开方法来研究与格树和格动物有关的一些问题，并研究渗流理论中出现的一种特殊随机环境中的随机游动。

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. The main thrust of this proposal is to further develop a new and flexible mathematically rigorous approach to the renormalisation group, and to apply it to obtain detailed information about 4-dimensional self-avoiding walks. A secondary thrust of this proposal is to further develop and apply the lace expansion method to some related problems concerning lattice trees and lattice animals, and to study random walk in a particular random environment arising in percolation theory.