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中的重要课题，在数学内外都具有重要的意义。他们的分析需要理解无限数量的随机变量，这些随机变量的相互依赖性强到足以导致长期相关性。寻找新的数学方法来描述这种情况是概率论的前沿，也是本提案的主题。******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避免行走模型。研究简化数学模型的一个很好的理由是临界现象具有普遍性：宏观的临界行为被预测为独立于模型如何在微观上定义的细节。通用性允许从初看起来过于简化的模型中得出物理上有意义的结论。通用临界行为的一个重要方面包含在临界指数中，它控制着临界点附近的相关长度和磁化率等量。******自旋和自我避免行走模型都可以在任何维度d的晶格上定义。申请人和他的合作者最近开发了一种数学上严格且通用的重整化群（RG）方法，用于分析维度d=4。本建议的主要目的是将RG方法扩展到分析4维以下的关键现象。为了理解较低的维度，自20世纪70年代早期威尔逊和费雪关于ε展开的开创性工作以来，分数维度已经在物理文献中用于临界指数的计算。分数维的使用在数学上是有问题的，但是另一种方法是通过考虑基于拉普拉斯算子分数次幂的远程模型来模拟分数维的使用。该建议的主要目的是扩展RG方法，以获得d=1,2,3维的epsilon展开的数学严格版本，用于此类远程模型。这将在欧几里得格上提供第一个这类结果。然后，该方法将用于证明其他关键指数的存在性和计算其他关键指数的值，用于相关长度和各种关键相关函数（包括两点函数），并研究临界尺度极限和通用性。长期目标是应用该方法分析自避步行的端到端距离。这项研究将为重要的物理问题提供严格的数学基础，这些问题的数学理解在40多年来一直没有得到解决。

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.