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SUMMARY:Random Surfaces and Quantum Loewner Evolution - Jason Miller (MIT)
DTSTART:20140130T131500Z
DTEND:20140130T140000Z
UID:TALK50589@talks.cam.ac.uk
CONTACT:HoD Secretary\, DPMMS
DESCRIPTION:What is the canonical way to choose a random\, discrete\, two-
 dimensional manifold which is homeomorphic to the sphere?  One procedure f
 or doing so is to choose uniformly among the set of surfaces which can be 
 generated by gluing together $n$ Euclidean squares along their boundary se
 gments in such a way that the resulting surface is homeomorphic to the sph
 ere.  This is an example of what is called a random planar map and is a mo
 del of what is known as pure discrete quantum gravity.  The asymptotic beh
 avior of these discrete\, random surfaces has been the focus of a large bo
 dy of literature in both probability and combinatorics.  This has culminat
 ed 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 ra
 ndom metric space is called the Brownian map.  \n\nAnother canonical way t
 o choose a random\, two-dimensional manifold is what is known as Liouville
  quantum gravity (LQG).  This is a theory of continuum quantum gravity int
 roduced by Polyakov to model the time-space trajectory of a string.  Its m
 etric when parameterized by isothermal coordinates is formally described b
 y $e^{\\gamma h}      \n(dx^{2} + dy^2)$ where $h$ is an instance of the c
 ontinuum Gaussian free field\, the standard Gaussian with respect to the D
 irichlet inner product.  Although $h$ is not a function\, Duplantier and S
 heffield succeeded in constructing LQG rigorously as a random area measure
 .  LQG for $\\gamma=\\sqrt{8/3}$ is conjecturally equivalent to the Browni
 an map and to the limits of other discrete theories of quantum gravity for
  other values of $\\gamma$.\n\nIn this talk\, I will describe a new family
  of growth processes called quantum Loewner evolution (QLE) which we propo
 se using to endow LQG with a distance function which is isometric to the B
 rownian map.  I will also explain how QLE is related to DLA\, the dielectr
 ic breakdown model\, and SLE.\n\nBased on joint works with Scott Sheffield
 .
LOCATION:CMS\, MR5
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