Abstract
Liouville quantum gravity (LQG) surfaces are a family of random fractal
surfaces which can be thought of as the canonical models of random
two-dimensional Riemannian manifolds, in the same sense that Brownian motion is
the canonical model of a random path. LQG surfaces are the continuum limits of
discrete random surfaces called random planar maps. In this expository article,
we discuss the definition of random planar maps and LQG, the sense in which the
former converges to the latter, and the motivations for studying these objects.
We also mention several open problems. We do not assume any background
knowledge beyond that of a second-year mathematics graduate student.
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