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）我们已经开发了一种形式主义，可以在拓扑磁盘上重写二维指标，恒定曲率，作为一个随机场（具有特定电位），本身对应于自相传的曲线。已经研究了这些曲线的许多组合特性（平均长度，面积，等值不平等，陀螺半径，绕组数，绕组数，自相隔）。这样的随机物体在布朗运动和自我避开曲线之间提供了一个中间的非马克维亚随机过程，这些过程与二维量子重力具有巨大相关性，并通过SYK/JT对应关系与黑洞的连接，这些曲线为这些曲线提供了这些曲线，这些曲线为算法提供了一个随机的步行过程。操作员的组合解释（我们将其应用于Galton-Watson树图和Bethe晶格）；通过对时间传播器的生成函数的研究，我们获得了它们的热内核，对助行器的光谱维度进行了粗略的估计（这似乎与标准的随机助行器没有区别，即4/3和3）；为了利用用于生成尖的图（尤其是树木和平面图）功能的通用关系，我们了解到，在计算图形上n-点相关函数时，正是它们的渐近学（接近其奇异性）。

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