Universality and conformal invariance of certain two dimensional statistical physics models

某些二维统计物理模型的普遍性和共形不变性

• 批准号: RGPIN-2018-04122
• 负责人: Ray, Gourab
• 金额: $ 1.68万
• 依托单位: University of Victoria
• 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=712749
• 关键词: Universality; conformal; invariance; certain; two

My research is in the field of probability theory primarily focussing on problems arising from the field of statistical and mathematical physics. Statistical physics tries to justify various physical phenomena (for example phase transition like melting of ice, demagnetization of magnets at high temperature etc) using a model which is inherently probabilistic in nature. The basic idea is to provide a framework to describe a macroscopic phenomenon (like transition of state, which we can actually see in everyday life) as an outcome of the microscopic behaviour of the molecules and atoms. The premise is that the microscopic behaviour is random which can be described using a probabilistic model, and the large scale statistical behaviour of this model is precisely what we observe as a macroscopic phenomenon. My research is based on problems related to the universal large scale behaviours of such statistical physics models. The questions become particularly interesting at or near criticality, that is, at or near the precise point when the phase transition occurs. During and around my PhD, I worked on problems built into a framework where the underlying geometry is also random, which leads to the topic of Liouville quantum gravity and random planar maps. Recently I am also interested in gradient models, particularly the dimer model, random homomorphisms and the six vertex model. Generally, one can think of these models as models of random functions and in the planar case, these functions can be viewed as a random surface. My focus is mainly in the planar case, where I am working on toy problems around several big conjectures regarding the universal fluctuations of these random surfaces. My current research program consists primarily of two such models. One is concerned broadly with establishing universality of fluctuations in the dimer model (which is a model of random perfect matchings on graphs). With my collaborators in Cambridge and Paris, we have developed a novel technique which enables us to obtain a universality result. At the moment, we are writing a series of papers on problems along these lines, a couple of which are already submitted. The second program involves random homomorphisms from the square lattice to integers. Along with my collaborators in Paris, we are trying to develop a renormalization technique to obtain information about the correlations of these models. Finding out precise information about these models is quite challenging and many questions have been open for a while. Our approach is to utilize the renormalization techniques recently developed to analyze a similar model. These two programs give rise to several projects and sub-projects. In the upcoming years, my target would be to make progress on these questions of interest.

我的研究是在概率论领域，主要集中在统计和数学物理领域产生的问题。统计物理试图证明各种物理现象（例如相变，如冰的融化，高温下磁铁的退磁等）使用一个模型，这是本质上固有的概率。其基本思想是提供一个框架来描述宏观现象（如我们在日常生活中实际看到的状态转变），作为分子和原子微观行为的结果。前提是微观行为是随机的，可以用概率模型来描述，而这个模型的大尺度统计行为正是我们观察到的宏观现象。我的研究是基于与这种统计物理模型的普遍大尺度行为相关的问题。这些问题在临界或接近临界时变得特别有趣，也就是说，在相变发生的精确点或附近。 在我的博士学位期间和前后，我致力于构建一个框架的问题，其中底层几何也是随机的，这导致了刘维尔量子引力和随机平面映射的主题。最近我也对梯度模型很感兴趣，特别是二聚体模型，随机同态和六顶点模型。一般来说，人们可以认为这些模型是随机函数的模型，在平面的情况下，这些函数可以被视为一个随机表面。我的重点主要是在平面的情况下，在那里我工作的玩具问题，围绕几个大的几何图形有关的普遍波动，这些随机表面。 我目前的研究计划主要包括两个这样的模型。一个是广泛关注建立普遍性的波动在二聚体模型（这是一个模型的随机完美匹配的图）。我们与剑桥和巴黎的合作者一起开发了一种新技术，使我们能够获得普遍性结果。目前，我们正在沿着这些思路撰写一系列关于问题的论文，其中几篇已经提交。 第二个程序涉及从正方形格到整数的随机同态。沿着我在巴黎的合作者们，我们正试图发展一种重整化技术，以获得关于这些模型的相关性的信息。找出关于这些模型的精确信息是相当具有挑战性的，许多问题已经开放了一段时间。我们的方法是利用最近发展的重整化技术来分析一个类似的模型。 这两个方案产生了若干项目和分项目。在今后几年里，我的目标是在这些令人感兴趣的问题上取得进展。