Liouville quantum gravity and conformal probability

刘维尔量子引力是一个产生随机二维表面的模型。该模型是波利亚科夫30年前在弦理论的背景下提出的，现在是物理学的一个分支--保形场理论的核心。刘维尔量子引力是一个连续体理论，但它有离散的类似物。就像有离散的方法来创建随机路径(通过在网格上绘制它们)一样，也有离散的方法来产生随机曲面(通过将相同的单位正方形沿其边缘粘合在一起)。PI将研究离散和连续随机曲面：它们之间的关系，它们与规范理论和理论物理其他方面的关系，以及它们与保角随机几何和随机曲线(如Schramm-Loewner演化)的关系。特别是，PI旨在表明，在某种意义上，如果一个人从大量非常小的正方形中建立一个离散的随机表面，它看起来非常像在Liouville量子引力中研究的连续随机表面。此外，当附加的统计物理结构(Ising或Potts模型)与离散模型相关联时，该结构对应于连续随机表面上的连续随机几何对象(保形环系综)。PI将与本科生、研究生和博士后研究员在概率、分析和统计物理中的这些问题和相关问题上密切合作。

Liouville quantum gravity is a model for producing a random two-dimensional surface. The model was introduced by Polyakov 30 years ago in the context of string theory and is now a centerpiece of a branch of physics called conformal field theory. Liouville quantum gravity is a continuum theory, but it has discrete analogs. Just as there are discrete ways to create random paths (by drawing them on grids) there are also discrete ways to produce random surfaces (by gluing together identical unit squares along their edges). The PI will study both discrete and continuum random surfaces: their relationship to each other, their relationship to gauge theories and other aspects of theoretical physics, and their relationship to conformal random geometry and random curves such as the Schramm-Loewner evolution. In particular, the PI aims to show that, in some sense, if one builds a discrete random surface out of a large number of very small squares, it looks very much like the continuum random surfaces studied in Liouville quantum gravity. Moreover, when additional statistical physical structure (Ising or Potts models) is associated with the discrete models, this structure corresponds to continuum random geometric objects (conformal loop ensemble) on the continuum random surfaces. The PI will work closely with undergraduate students, graduate students, and postdoctoral fellows, on these and related problems in probability, analysis, and statistical physics.