Quantum fields and random geometries

量子场和随机几何形状

• 批准号: 21K20340
• 负责人: DELPORTE Nicolas
• 金额: $ 1.83万
• 依托单位: Okinawa Institute of Science and Technology Graduate University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Research Activity Start-up
• 财政年份: 2021
• 资助国家: 日本
• 起止时间: 2021-08-30 至 2024-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-21K20340/
• 关键词: Random walk; Self overlapping curve; Dirac operator; Random Graph; long-range; fermions; Random geometry; Quantum field theory; Renormalization

1) We have developed a formalism to rewrite a partition function over two-dimensional metrics on topological disks, of constant curvature, as a random field (with a specific potential), that itself corresponds to self-overlapping curves. Many combinatorial properties of those curves have been studied (average length, area, isoperimetric inequality, gyroscopic radius, winding number, self-intersections). Such a random object provides an intermediate non-Markovian random process between the Brownian motion and the self-avoiding curve, that has tremendous relevance for two dimensional quantum gravity with connections to black holes through the SYK/JT correspondence, for which those curves provide the exact degrees of freedom to comprehend.2) We have implemented a random walk process generated by an operator corresponding to the square root of a graph Laplacian, giving to the inverse of that operator a combinatorial interpretation (we have applied it to Galton-Watson tree graphs and the Bethe lattice); Through the study of the generating function for the time propagator, we have obtained their heat-kernel, giving a crude estimate of the spectral dimension of the walker (that seem not to differ from the standard random walker, ie 4/3 and 3 respectively) ; In order to exploit general relations obtained for generating functions of pointed graphs (especially trees and planar maps), we understood that it is their asymptotics (close to their singularities) that matter in the computation of n-point correlation functions of fields on the graphs.

1）我们开发了一种形式主义，将曲率恒定的拓扑盘上的二维度量重写为随机场（具有特定势），其本身对应于自重叠曲线。人们已经研究了这些曲线的许多组合特性（平均长度、面积、等周不等式、陀螺半径、缠绕数、自交点）。这样的随机物体提供了布朗运动和自回避曲线之间的中间非马尔可夫随机过程，该过程与通过 SYK/JT 对应关系与黑洞连接的二维量子引力具有巨大的相关性，为此这些曲线提供了2）我们已经实现了由对应于图拉普拉斯算子的平方根的算子生成的随机游走过程，为该算子的逆提供了组合解释（我们已将其应用于Galton-Watson 树图和 Bethe 格）；通过研究时间传播器的生成函数，我们获得了它们的热核，粗略地估计了步行者的谱维数（这似乎与标准随机步行者没有什么不同，即分别为 4/3 和 3） ）；为了利用生成点图（尤其是树和平面图）函数所获得的一般关系，我们知道它们的渐近性（接近奇点）在计算图上场的 n 点相关函数时很重要。

项目成果

A random walk approach to two dimensional quantum gravity

二维量子引力的随机游走方法

• 发表时间: 2023
• 作者: Nicolas Delporte

Peeking at quantum gravity with self-overlapping curves

通过自重叠曲线观察量子引力

• 发表时间: 2023
• 作者: Y. Goto;M. Taniguchi;Nicolas Delporte
• 通讯作者: Nicolas Delporte

On aspects of two-dimensional quantum gravity

关于二维量子引力的各个方面

• 发表时间: 2023
• 作者: Y. Goto;Suzuki. K.;Xu;X.;and M. Taniguchi;只野之英;Nicolas Delporte
• 通讯作者: Nicolas Delporte