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CATEGORIES:Special DPMMS Colloquium
SUMMARY:Random Surfaces and Quantum Loewner Evolution - Ja
son Miller (MIT)
DTSTART;TZID=Europe/London:20140130T131500
DTEND;TZID=Europe/London:20140130T140000
UID:TALK50589AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/50589
DESCRIPTION:What is the canonical way to choose a random\, dis
crete\, two-dimensional manifold which is homeomor
phic to the sphere? One procedure for doing so is
to choose uniformly among the set of surfaces whi
ch can be generated by gluing together $n$ Euclide
an squares along their boundary segments in such a
way that the resulting surface is homeomorphic to
the sphere. This is an example of what is called
a random planar map and is a model of what is kno
wn as pure discrete quantum gravity. The asymptot
ic behavior of these discrete\, random surfaces ha
s been the focus of a large body of literature in
both probability and combinatorics. This has culm
inated with the recent works of Le Gall and Miermo
nt which prove that the $n \\to \\infty$ distribut
ional limit of these surfaces exists with respect
to the Gromov-Hausdorff metric after appropriate r
escaling. The limiting random metric space is cal
led the Brownian map. \n\nAnother canonical way t
o choose a random\, two-dimensional manifold is wh
at is known as Liouville quantum gravity (LQG). T
his is a theory of continuum quantum gravity intro
duced by Polyakov to model the time-space trajecto
ry of a string. Its metric when parameterized by
isothermal coordinates is formally described by $e
^{\\gamma h} \n(dx^{2} + dy^2)$ where $h$ is
an instance of the continuum Gaussian free field\,
the standard Gaussian with respect to the Dirichl
et inner product. Although $h$ is not a function\
, Duplantier and Sheffield succeeded in constructi
ng LQG rigorously as a random area measure. LQG f
or $\\gamma=\\sqrt{8/3}$ is conjecturally equivale
nt to the Brownian map and to the limits of other
discrete theories of quantum gravity for other val
ues of $\\gamma$.\n\nIn this talk\, I will describ
e a new family of growth processes called quantum
Loewner evolution (QLE) which we propose using to
endow LQG with a distance function which is isomet
ric to the Brownian map. I will also explain how
QLE is related to DLA\, the dielectric breakdown m
odel\, and SLE.\n\nBased on joint works with Scott
Sheffield.
LOCATION:CMS\, MR5
CONTACT:HoD Secretary\, DPMMS
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