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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=679461
• 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.**

我的研究领域是概率论，主要集中在统计和数学物理领域产生的问题。统计物理学试图用一个本质上具有概率性的模型来证明各种物理现象（例如，像冰融化这样的相变，高温下磁铁的退磁等）。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可以看到的状态转变），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所观察到的宏观现象。我的研究是基于与这种统计物理模型的普遍大规模行为相关的问题。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。******在我读博士期间和前后，我研究了一个框架内的问题，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图的主题。最近我对梯度模型也很感兴趣，特别是二聚体模型，随机同态和六顶点模型。一般来说，我们可以把这些模型看作是随机函数的模型，在平面的情况下，这些函数可以看作是一个随机曲面。我的重点主要是在平面的情况下，我正在研究一些关于这些随机表面的普遍波动的大猜想的玩具问题。******我目前的研究计划主要包括两个这样的模型。一个是在二聚体模型（图上的随机完美匹配模型）中建立波动的普适性。与我在剑桥和巴黎的合作者一起，我们开发了一种新技术，使我们能够获得普适性结果。目前，我们正在写一系列关于这些问题的论文，其中一些已经提交。******第二个程序涉及从方形晶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归一化技术，以获取这些模型之间相关性的信息。找到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开发的重整化技术来分析类似的模型。******这两个方案产生了几个项目和子项目。在接下来的几年里，我的目标是在这些感兴趣的问题上取得进展