Renormalisation group and critical phenomena

重正化群和临界现象

• 批准号: RGPIN-2017-03748
• 负责人: Slade, Gordon
• 金额: $ 3.13万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=710408
• 关键词: Renormalisation; group; critical; phenomena

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.

相变和临界现象的数学模型是统计力学和概率论中的主要课题，在数学内外都有重要的意义。他们的分析需要理解无限多的随机变量，这些变量的相互依赖性足够强，足以导致长期相关性。寻找新的数学来描述这种情况是概率论的前沿，也是本提案的主题。 我们将研究临界现象的两个基本模型：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：宏观临界行为被预测为与模型如何在微观上定义的细节无关。普遍性允许从乍一看似乎过于简化的模型中得出有实际意义的结论。 普遍的临界行为的一个重要方面是由临界指数，如相关长度和磁化率附近的临界点的数量。 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一种数学上严格和通用版本的重正化群（RG）方法，以分析维度d=4。 这个建议的主要目的是扩展这个RG方法来分析4维以下的临界现象。 为了理解更低的维度，分数维在物理学文献中被用于计算临界指数，自从威尔逊和费舍尔在1970年代早期对π展开的开创性工作以来。 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程模型来模拟分数维的使用。 这个建议的主要目的是为了扩展RG方法，以获得在d= 1，2，3维中，对于这样的长距离模型的严格的数学版本。这将提供第一个结果，这种对欧几里德格。 然后，该方法将被用来证明存在和计算其他临界指数的值，为相关长度和各种关键的相关功能，包括两点功能，并研究临界标度限制和普适性。一个长期的目标是应用这些方法来分析自我回避行走的端到端距离。 这项研究将为物理学感兴趣的重要问题提供严格的数学基础，这些问题的数学理解已经解决了40多年。