Gaussian Free Field and Conformal Loop Ensemble

这项研究是在概率和统计物理领域。统计物理学家和概率学家经常试图理解由许多微观随机输入组成的系统的宏观行为，这些输入可以在临界温度下产生两相之间的界面，例如零摄氏度的水和冰。这可以通过关键情况下离散晶格模型(微观输入)的尺度极限行为(宏观行为)来模拟。Oded Schramm的SLE(随机洛夫纳进化)过程使数学家和物理学家对二维离散模型中界面的比例极限有了全新的理解。而CLE(Conform Loop EnSemble)是SLE的推广，被预测为离散模型中所有界面集合的尺度极限。本文主要研究共形不变度量及其与高斯自由场、Liouville量子引力和随机映象之间的关系。自从SLE和CLE的引入以来，CLE(4)已被证明是离散高斯自由场能级线集合的标度极限。在前人的工作中，构造了CLE(4)环组态的时间参数，并给出了高斯自由场与CLE(4)之间随时间参数的耦合。研究的目的是证明CLE(4)上定义的时间参数实际上是环路构型的确定性函数。这一结果将加深对高斯自由场与CLE之间关系的理解。(2)CLE修饰的Liouville量子引力。刘维尔量子引力被认为是随机映射的标度极限。该研究项目首先旨在了解CLE装饰的Liouville量子引力，然后探索CLE装饰的Liouville量子引力与回路装饰的随机映射之间的关系。

This research is in the area of Probability and Statistical Physics. Statistical physicists and probabilists often try to understand the macroscopic behavior of systems consisting of many microscopic random inputs, which can give rise to interfaces between two phases at a critical temperature, such as water and ice at zero degree Celsius. This can be modeled via the scaling limit behavior (macroscopic behavior) of discrete lattice models (microscopic inputs) in critical cases. Oded Schramm's SLE (Stochastic Loewner Evolution) processes have led mathematicians and physicists to a clean and novel understanding of the scaling limits of the interfaces in discrete models in two dimensions. And CLE (Conformal Loop Ensemble) is the generalization of SLE which is predicted to be the scaling limit of the collection of all interfaces in discrete models. This research focuses on CLE and its relation between Gaussian Free Field, Liouville Quantum Gravity, and Random Maps.Precisely, the research considers the following two problems.(1) The conformally invariant metric on CLE. Since the introduction of SLE and CLE, CLE(4) has been proved to be the scaling limit of the collection of level lines of discrete Gaussian Free Field. In the previous work of the principal investigator, a time parameter is constructed for CLE(4) loop configurations, and a coupling between Gaussian Free Field and CLE(4) with time parameter is given. The research aims to show that the time parameter defined on CLE(4) is in fact a deterministic function of the loop configuration. This result would deepen understanding of the relation between Gaussian Free Field and CLE.(2) CLE-decorated Liouville Quantum Gravity. Liouville Quantum Gravity is conjectured to be the scaling limit of random maps. The research project first aims to understand CLE-decorated Liouville Quantum Gravity and then to explore the relation between CLE-decorated Liouville Quantum Gravity and loop-decorated random maps.