Renormalisation group and critical phenomena

相变和临界现象的数学模型是统计力学和概率论的主要课题，在数学内部和外部都有重要的意义。他们的分析需要了解无限数量的随机变量，这些变量的相互依存性足够强，足以引起长期相关性。寻找新的数学来描述这一情景是概率论的前沿，也是这个提议的主题。*将研究两个临界现象的基本模型：铁磁性的晶格自旋模型和线性聚合物的自回避行走模型。研究简化数学模型的一个很好的理由是，临界现象表现出普遍性：宏观临界行为被预测为独立于模型如何在微观上定义的精细细节。普遍性允许从乍一看似乎过于简化的模型中得出物理上有意义的结论。普遍临界行为的一个重要方面被临界指数所包含，临界指数控制着临界点附近的关联长度和磁化率等量。自旋和自回避行走模型都可以定义在任意维d的晶格上。申请人和他的合作者最近开发了数学上严格的通用版本的重整化群(RG)方法，以分析维d=4。该建议的主要目的是将该RG方法扩展到分析低于4维的临界现象。为了理解较低的维，在物理文献中使用分数维来计算临界指数，自20世纪70年代初威尔逊和费雪在爱西隆膨胀方面的开创性工作以来。分数维的使用在数学上是有问题的，但另一种方法是通过考虑基于拉普拉斯算子的分数幂的长期模型来模拟分数维的使用。这一建议的主要目的是扩展RG方法，以获得这样的长程模型的d=1，2，3维的epsilon展开的数学严谨版本。这将提供欧几里得格子上此类结果的第一个结果。然后，这些方法将被用来证明和计算关联长度和包括两点函数在内的各种临界关联函数的其他临界指数的值，并研究临界标度极限和普适性。一个长期的目标是应用这些方法来分析自我回避步行的端到端距离。这项研究将为40多年来一直悬而未决的物理兴趣重要问题提供严密的数学基础。

Mathematical models of phase transition and critical phenomena are major topics in statistical mechanics and probability theory, with important implications both within and outside mathematics. Their analysis requires the understanding of an infinite number of random variables whose mutual dependence is strong enough to cause long-range correlations. The search for new mathematics to describe this scenario is at the cutting edge of probability theory, and is the topic of this proposal.******Two fundamental models of critical phenomena will be studied: lattice spin models of ferromagnetism, and the self-avoiding walk model of linear polymers. A good reason for studying simplified mathematical models is that critical phenomena exhibit universality: the macroscopic critical behaviour is predicted to be independent of the fine details of how the model is defined microscopically. Universality allows physically meaningful conclusions to be drawn from models which at first sight may seem overly simplified. An important aspect of the universal critical behaviour is encompassed by the critical exponents that govern quantities such as correlation length and susceptibility near the critical point.******Both the spin and the self-avoiding walk models can be defined on a lattice of any dimension d. The applicant and his collaborators have recently developed a mathematically rigorous and general version of the renormalisation group (RG) method, to analyse dimension d=4. The primary thrust of this proposal is to extend this RG method to analyse critical phenomena below dimension 4. To understand lower dimensions, fractional dimensions have been used in the physics literature for the computation of critical exponents, since the pioneering work of Wilson and Fisher on the epsilon expansion in the early 1970s. The use of fractional dimensions is mathematically problematic, but an alternate approach is to mimic the use of fractional dimensions by considering long-range models based on a fractional power of the Laplace operator. The primary thrust of this proposal is to extend the RG method to obtain a mathematically rigorous version of the epsilon expansion in dimensions d=1,2,3, for such long-range models. This will provide the first results of this kind on a Euclidean lattice. The methods will then be used to prove existence of and compute the values of other critical exponents, for the correlation length and for various critical correlation functions including the two-point function, and to study critical scaling limits and universality. A long-term goal is to apply the methods to analyse the end-to-end distance for the self-avoiding walk. This research will provide a rigorous mathematical foundation for important problems of physical interest, whose mathematical understanding has remained unresolved for more than 40 years.