随机几何与量子时空：从标度不变随机几何和渐近安全到随机双曲曲面与JT引力
Random Geometry and Quantum Spacetime: From scale-invariant random geometries and asymptotic safety to random hyperbolic surfaces and JT gravity
AI总结：
本论文围绕随机几何对量子时空结构的启示展开，综述随机几何基础，用Mating of Trees构建标度不变随机几何以探索渐近安全紫外固定点，研究JT引力的随机几何表述并发现新形变，还深入探究离散几何与双曲曲面的关联。
摘要：
本论文围绕一个核心问题展开：“我们能从随机几何中了解到量子时空的哪些结构？”在第2章中，我们对本论文的数学基础——随机几何进行了部分综述。在第3章中，我们使用一种源自随机几何、名为Mating of Trees（树的配对）的构造方法，构建出标度不变的随机几何，这类几何出现在Liouville Quantum Gravity（刘维尔量子引力）中，且有望在二维和三维中实现Asymptotic Safety（渐近安全）所预言的紫外固定点。在第4章中，我们探究了JT引力的随机几何表述，以及我们对随机临界映射的理解如何带来了一类新的JT引力形变的发现。此外，JT引力与矩阵模型之间的关联促使我们在第5章中基于度量映射和不可约度量映射的几何，更深入地研究离散几何与双曲曲面之间的联系。
英文摘要：
This thesis is driven by a central question: "What can we learn from random geometries about the structure of quantum spacetime?" In Chapter 2, we provide a partial review of the mathematical foundation of this thesis, random geometry. In Chapter 3, we use a construction coming from random geometry called Mating of Trees to build scale-invariant random geometries that appear in Liouville Quantum Gravity and have the potential to implement the UV fixed point predicted by Asymptotic Safety in two and three dimensions. In Chapter 4 we explore the random geometry formulation of JT gravity and how our understanding of random critical maps yields the discovery of a new family of deformations of JT gravity. Furthermore, the connection between JT gravity and matrix models leads us to delve deeper into the link between discrete geometry and hyperbolic surfaces, building upon the geometry of metric maps and irreducible metric maps in Chapter 5.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
