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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)我们已经开发了一种形式主义重写的配分函数在二维度量拓扑磁盘上，常曲率，作为一个随机场（具有特定的潜力），这本身对应于自重叠曲线。这些曲线的许多组合性质已被研究（平均长度，面积，等周不等式，陀螺半径，缠绕数，自相交）。这样的随机对象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，这与通过SYK/JT对应与黑洞连接的二维量子引力具有巨大的相关性，这些曲线提供了精确的自由度，以使其保持不变。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行走过程，给出该算子的逆算子的组合解释，（我们已将其应用于Galton-Watson树图和Bethe格）;通过对时间传播子的生成函数的研究，得到了它们的热核，并对步行者的谱维数作了粗略的估计（这似乎没有区别于标准的随机步行者，即4/3和3分别）;为了利用点图生成函数的一般关系，（尤其是树和平面地图），我们知道它们的渐近性（接近它们的奇点），这在计算图上的场的n点相关函数中很重要。