An expository overview of the definition of Liouville quantum gravity surfaces, the three senses in which random planar maps converge to them, and the major open problems.
Gaussian free fields for mathematicians
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abstract
The d-dimensional Gaussian free field (GFF), also called the (Euclidean bosonic) massless free field, is a d-dimensional-time analog of Brownian motion. Just as Brownian motion is the limit of the simple random walk (when time and space are appropriately scaled), the GFF is the limit of many incrementally varying random functions on d-dimensional grids. We present an overview of the GFF and some of the properties that are useful in light of recent connections between the GFF and the Schramm-Loewner evolution.
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2019 1verdicts
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Random surfaces and Liouville quantum gravity
An expository overview of the definition of Liouville quantum gravity surfaces, the three senses in which random planar maps converge to them, and the major open problems.