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.

我的研究是在概率理论领域主要集中在统计和数学物理学领域引起的问题。统计物理学试图使用本质上本质上具有概率的模型来证明各种物理现象（例如冰的熔化，高温等在高温等上的磁体融化）。基本思想是提供一个框架来描述宏观现象（例如状态的过渡，我们可以在日常生活中实际看到），作为分子和原子的微观行为的结果。前提是微观行为是随机的，可以使用概率模型来描述，并且该模型的大规模统计行为正是我们认为是宏观现象。我的研究基于与此类统计物理模型的通用大规模行为有关的问题。这些问题在临界或接近临界时变得尤为有趣，即在发生相变发生的确切点或附近。 在我的博士学位期间和周围，我研究了一个框架内的问题，其中基础几何形状也随机，这导致了liouville量子重力和随机平面图的主题。最近，我也对梯度模型感兴趣，尤其是二聚体模型，随机同构和六个顶点模型。通常，人们可以将这些模型视为随机函数的模型，在平面情况下，可以将这些功能视为随机表面。我的重点主要是在平面案例上，在那里我围绕着关于这些随机表面的普遍波动的几个重大猜想的玩具问题。 我当前的研究计划主要由两个这样的模型组成。人们广泛关注在二聚体模型中建立波动的普遍性（这是图表上随机完美匹配的模型）。与我在剑桥和巴黎的合作者一起，我们开发了一种新颖的技术，使我们能够获得普遍性结果。目前，我们正在撰写有关这些问题问题的一系列论文，其中一些已经提交。 第二个程序涉及从方格到整数的随机同态。与我在巴黎的合作者一起，我们正在尝试开发一种重新归一化技术，以获取有关这些模型相关性的信息。找到有关这些模型的精确信息非常具有挑战性，许多问题已经开放了一段时间。我们的方法是利用最近开发的重新归一化技术来分析类似的模型。 这两个程序引起了几个项目和次项目。在接下来的几年中，我的目标是在这些感兴趣的问题上取得进展。