The critical behaviour of random systems

在统计力学、概率论和组合学中，自我避免行走和渗透等数学模型引起了相当大的兴趣。这些模型已经被研究了大约半个世纪，对数学家来说仍然是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比以往任何时候都更大，这门学科正在蓬勃发展。

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.