Quantum fields and random geometries

1)我们已经开发了一种形式来重写拓扑磁盘上二维度量的配分函数，曲率恒定，作为一个随机场（具有特定的势），它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、圈数、自交）。这样一个随机物体在布朗运动和自我回避曲线之间提供了一个中间的非马尔可夫随机过程，这对于通过SYK/JT对应与黑洞联系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理解自由度。2)我们实现了一个随机游走过程，这个随机游走过程是由一个对应于一个图拉普拉斯算子的平方根的算子生成的，给了这个算子的逆一个组合解释（我们已经把它应用到高尔顿-沃森树图和贝特格上）；通过对时间传播器的生成函数的研究，我们得到了它们的热核，给出了行走器的光谱维数的粗略估计（似乎与标准的随机行走器没有区别，分别是4/3和3）；为了利用生成点图（特别是树和平面图）函数所获得的一般关系，我们了解到在计算图上的n点域相关函数时，它们的渐近性（接近它们的奇点）是很重要的。

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.