Universality and conformal invariance of certain two dimensional statistical physics models

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中出现的问题。统计物理学试图使用一个本质上是概率的模型来证明各种物理现象(例如，冰的融化、高温下磁体的退磁等相变)的合理性。其基本思想是提供一个框架来描述作为分子和原子微观行为的结果的宏观现象(如我们实际上可以在日常生活中看到的状态转变)。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作为宏观现象观察到的。我的研究是基于与此类统计物理模型的普遍大规模行为相关的问题。这些问题在临界点或临界点附近变得特别有趣，也就是说，在相变发生的准确点或附近。 在我攻读博士学位期间及前后，我研究了一个框架中的问题，其中基础几何也是随机的，这就引出了刘维尔量子重力和随机平面地图的主题。最近我也对梯度模型感兴趣，特别是二聚体模型、随机同态和六顶点模型。通常，人们可以将这些模型视为随机函数的模型，而在平面情况下，这些函数可以被视为随机曲面。我的重点主要放在平面问题上，在那里，我围绕着关于这些随机表面的普遍涨落的几个大猜想来研究玩具问题。 我目前的研究计划主要由两个这样的模型组成。一个广泛地涉及在二聚体模型(这是图上的随机完美匹配的模型)中建立涨落的普适性。与我在剑桥和巴黎的合作者一起，我们开发了一种新的技术，使我们能够获得普遍的结果。目前，我们正在就这些问题撰写一系列论文，其中几篇已经提交。 第二个程序涉及从正方形格子到整数的随机同态。与我在巴黎的合作者一起，我们正试图开发一种重整化技术，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准确信息相当具有挑战性，许多问题已经悬而未决一段时间了。我们的方法是利用最近发展起来的重整化技术来分析类似的模型。 这两个项目产生了几个项目和子项目。在接下来的几年里，我的目标是在这些令人感兴趣的问题上取得进展。

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.