Two dimensional percolation and Liouville quantum gravity

Speaker 

孙鑫，2011年获得北京大学数学学士，2017年获得麻省理工学院数学博士，2020年起任宾夕法尼亚大学数学系助理教授，2023年9月起任北京国际数学研究中心长聘副教授，研究领域为概率论和数学物理，主要方向是随机几何、统计物理、和量子场论。荣获2023年度戴维逊（Rollo Davidson）奖。

Abstract

Smirnov's proof of Cardy's formula for percolation on the triangular lattice leads to a discrete approximation of conformal maps, which we call the Cardy-Smirnov embedding.

Under this embedding, Holden and I proved that the uniform triangulation converge to a continuum random geometry called pure Liouville quantum gravity. There is a variant of the Gaussian free field governing the random geometry, which is an important example of conformal field theory called Liouville CFT. A key motivation for understanding Liouville quantum gravity rigorously is its application to the evaluation of scaling exponents and dimensions for 2D critical systems such as percolation.

Recently, with Nolin, Qian and Zhuang, we used this idea and the integrable structure of Liouville CFT to derive a scaling exponent for planar percolation called the backbone exponent, which was unknown for several decades.