**Joint Probability and Rainwater seminar**

** Time: ** Tuesday, April 22, 2014, 1:30-3:30 pm.

** Location: ** SMI 307

** Speaker:** Jason Miller (MIT)

** Title:** Random Surfaces and Quantum Loewner Evolution

** Abstract: **
What is the canonical way to choose a random, discrete, two-dimensional manifold
which is homeomorphic to the sphere? One procedure for doing so is to choose uniformly
among the set of surfaces which can be generated by gluing together $n$ Euclidean squares
along their boundary segments. This is an example of what is called a random planar map
and is a model of what is known as pure discrete quantum gravity.
The asymptotic behavior of these discrete, random surfaces has been the focus of
a large body of literature in both probability and combinatorics.
This has culminated with the recent works of Le Gall and Miermont which prove that
the $n \to \infty$ distributional limit of these surfaces exists with respect to
the Gromov-Hausdorff metric after appropriate rescaling.
The limiting random metric space is called the Brownian map.

Another canonical way to choose a random, two-dimensional manifold is what is known
as Liouville quantum gravity (LQG). This is a theory of continuum quantum gravity
introduced by Polyakov to model the time-space trajectory of a string.
Its metric when parameterized by isothermal coordinates is formally described
by $e^{\gamma h} (dx^2 + dy^2)$ where $h$ is an instance of the continuum Gaussian free
field, the standard Gaussian with respect to the Dirichlet inner product.
Although $h$ is not a function, Duplantier and Sheffield succeeded in constructing
LQG rigorously as a random area measure. LQG for $\gamma=\sqrt{8/3}$ is conjecturally
equivalent to the Brownian map and to the limits of other discrete theories of
quantum gravity for other values of $\gamma$.

In this talk, I will describe a new family of growth processes called
quantum Loewner evolution (QLE) which we propose using to endow LQG with a distance
function which is isometric to the Brownian map. I will also explain how QLE is related
to DLA, the dielectric breakdown model, and SLE.

Based on joint works with Scott Sheffield

Archive of previous talks

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Mathematics
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University of
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