Random Walks and Scaling Limits

[405021] lawler在过去的几年中，在统计物理学中对二维晶格模型“临界”的严格理解方面取得了重大进展。许多连续模型，特别是Schramm-Loewner演化（SLE），已经被构造，并且在某些情况下已经证明离散模型在极限上接近SLE。提议者将研究一些离散模型，如自我回避行走、拉普拉斯随机行走、特定随机图上的行走等，希望表明它们也收敛于SLE。此外，提议者将考虑三维模型，其中最近开发的技术依赖于保形不变性不适用。目标是找到三维连续模型作为离散系统极限的候选者。本提案的目标是构建和分析相变的数学模型，相变是研究物理系统在改变参数时的急剧变化，如温度降低时水的冻结。更一般地说，数学的目标是理解允许人们从微观规则预测宏观行为的普遍性原则。这些结果不仅对概率论很重要，而且与理论物理的许多领域有关。特别的重点将放在二维模型的极限方法上，其中极限本身现在已经很好地理解了，并在问题更具挑战性的三维模型中构建极限的候选者。

0405021Lawler There has been significant progress in the last few years in the rigorous understanding of two-dimensional lattice models in statistical physics "at criticality". A number of continuous models, most particularly the Schramm-Loewner evolution (SLE), have been constructed, and in some cases it has been proved that discrete models approach SLE in the limit. The proposer will study a number of discrete models, e.g., self-avoiding walk, Laplacian random walks, and walks on certain random graphs, with the hope of showing that they also converge to SLE. Also, the proposer will consider models in three dimensions where the recently developed techniques which rely on conformal invariance to not apply. The goal is to find three-dimensional continuous models to be candidates for limits of discrete systems. The goal of this proposal is to construct and analyze mathematical models for phase transition, which is the study of the sharp changes in a physical system when changing a parameter such as the freezing of water when the temperature is reduced. More generally, the mathematical goal is to understand universality principles that allow one to predict macroscopic behavior from microscopic rules. As well as being important to probability theory, the results will be relevant to many areas of theoretical physics. Special focus will be placed on the approach to the limit for two-dimensional models where the limit itself is now well understood and to construct candidates for the limit in three dimensions where the problems are more challenging.