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CATEGORIES:Probability
SUMMARY:Existence and uniqueness of the Liouville quantum
gravity metric for γ ∈ (0\, 2) - Ewain Gwynne (Cam
bridge)
DTSTART;TZID=Europe/London:20190521T140000
DTEND;TZID=Europe/London:20190521T150000
UID:TALK123829AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/123829
DESCRIPTION:We show that for each $\\gamma \\in (0\,2)$\, ther
e is a unique metric associated with $\\gamma$-Lio
uville quantum gravity (LQG). More precisely\, we
show that for the whole-plane Gaussian free field
(GFF) $h$\, there is a unique random metric $D_h
= ``e^{\\gamma h} (dx^2 + dy^2)"$ on $\\mathbb C$
which is characterized by a certain list of axioms
: it is locally determined by $h$ and it transform
s appropriately when either adding a continuous fu
nction to $h$ or applying a conformal automorphism
of $\\BB C$ (i.e.\, a complex affine transformati
on). Metrics associated with other variants of the
GFF can be constructed using local absolute conti
nuity.\n\nThe $\\gamma$-LQG metric can be construc
ted explicitly as the scaling limit of \\emph{Liou
ville first passage percolation} (LFPP)\, the rand
om metric obtained by exponentiating a mollified v
ersion of the GFF. Earlier work by Ding\, Dub\\'ed
at\, Dunlap\, and Falconet (2019) showed that LFPP
admits non-trivial subsequential limits. We show
that the subsequential limit is unique and satisfi
es our list of axioms. In the case when $\\gamma =
\\sqrt{8/3}$\, our metric coincides with the $\\s
qrt{8/3}$-LQG metric constructed in previous work
by Miller and Sheffield\, which in turn is equival
ent to the Brownian map for a certain variant of t
he GFF. For general $\\gamma \\in (0\,2)$\, we con
jecture that our metric is the Gromov-Hausdorff li
mit of appropriate weighted random planar map mode
ls\, equipped with their graph distance. \n\nBase
d on four joint papers with Jason Miller and one j
oint paper with Julien Dub\\'edat\, Hugo Falconet\
, Josh Pfeffer\, and Xin Sun.\n
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0W
B
CONTACT:Perla Sousi
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