Random Planar Geometry

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A central theme in probability theory is to understand large discrete models and their scaling limits. The limiting objects, which are usually characterized by certain symmetries and spatial independence, capture the universal large-scale behavior of many models. In the last few decades, there have been great advances in random planar geometry, especially in the understanding of some fundamental two-dimensional discrete models and their scaling limits. These developments hugely expand our knowledge on the randomness of basic objects such as curves, functions, trees, and surfaces. They also revolutionize the mathematical understanding of some central pieces of physics, including conformal field theory, critical phenomena, and quantum gravity. This project aims to broaden understanding of the fundamental mathematical underpinnings in this subject.The project will explore two research directions in random planar geometry. In the first direction, the project aims at linking two ways of constructing random surfaces: (1) through the scaling limits of random planar maps; (2) through a continuous theory called the Liouville quantum gravity (LQG). The investigator plans to prove a conjecture asserting that LQG is the scaling limit of random planar maps in a strong sense. As a tool for proving this conjecture, the project considers a statistical mechanical model called the critical percolation and aims to establish that the scaling limit of the critical percolation on the uniform random triangulation and on a regular triangular lattice are the same. The ingredients in both investigations include: (a) combinatorial bijections for planar maps that encode their geometric information; (b) a relation between fractals in quantum and Euclidean geometry called the Knizhnik-Polyakov-Zamolodchikov relation; (c) the Fourier analysis of Boolean functions. In the second direction, the investigator aims to resolve questions in the geometry of random planar maps, computational geometry, and fractal geometry. The common theme in the approaches is the application of two recently-developed machineries in continuous random planar geometry called imaginary geometry and mating of trees.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.