Gaussian Free Field and Conformal Loop Ensemble

这项研究是在概率和统计物理学领域。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，例如在零摄氏度下的水和冰。这可以通过临界情况下离散晶格模型（微观输入）的标度极限行为（宏观行为）来建模。Oded Schramm的SLE（Stochastic Loewner Evolution）过程使数学家和物理学家对二维离散模型中界面的标度极限有了清晰而新颖的理解。共形环包络（CLE）是SLE的推广，SLE被认为是离散模型中所有界面集合的标度极限。本研究的重点是CLE及其与高斯自由场、刘维尔量子引力和随机映射之间的关系。(1)CLE上的共形不变度量。自SLE和CLE提出以来，CLE（4）已被证明是离散高斯自由场能级线集合的标度极限。在前期工作中，我们构造了CLE（4）圈的时间参数，并给出了高斯自由场与带时间参数的CLE（4）圈的耦合。研究的目的是表明CLE（4）中定义的时间参数实际上是环路结构的确定性函数。这一结果将加深对高斯自由场与CLE之间关系的理解。(2)刘维量子引力奖证明了Liouville量子引力是随机映射的标度极限。本研究的目的首先是了解CLE修饰的刘维尔量子引力，然后探讨CLE修饰的刘维尔量子引力与环修饰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.