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Quantum fields and random geometries

量子场和随机几何形状

基本信息

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

项目摘要

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)我们发展了一种形式，将具有常曲率的拓扑盘上的二维度量上的配分函数重写为(具有特定位势的)随机场，其本身对应于自重叠曲线。研究了这些曲线的许多组合性质(平均长度、面积、等周不等式、陀螺半径、缠绕数、自交)。这样的随机物体在布朗运动和自回避曲线之间提供了一个中间的非马尔可夫随机过程，对于通过SYK/JT对应与黑洞连接的二维量子引力具有巨大的关联，这些曲线提供了准确的理解自由度。2)我们实现了一个由对应于图的拉普拉斯平方根的算子产生的随机行走过程，给出了该算子的逆的组合解释(我们已经将其应用于Galton-Watson树图和Bethe格子)；通过对时间传播子母函数的研究，我们得到了时间传播子的热核，给出了步行子的谱维度的粗略估计(分别为4/3和3)；为了利用所得到的关于有点图(特别是树和平面映射)的母函数的一般关系，我们理解在图上的场的n点关联函数的计算中，是它们的渐近性(接近于它们的奇点)是重要的。