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SUMMARY:Existence and uniqueness of the Liouville quantum gravity metric f
 or γ ∈ (0\, 2) - Ewain Gwynne (Cambridge)
DTSTART:20190521T130000Z
DTEND:20190521T140000Z
UID:TALK123829@talks.cam.ac.uk
CONTACT:Perla Sousi
DESCRIPTION:We show that for each $\\gamma \\in (0\,2)$\, there is a uniqu
 e metric associated with $\\gamma$-Liouville 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 transforms appropriately when either addi
 ng a continuous function to $h$ or applying a conformal automorphism of $\
 \BB C$ (i.e.\, a complex affine transformation). Metrics associated with o
 ther variants of the GFF can be constructed using local absolute continuit
 y.\n\nThe $\\gamma$-LQG metric can be constructed explicitly as the scalin
 g limit of \\emph{Liouville first passage percolation} (LFPP)\, the random
  metric obtained by exponentiating a mollified version of the GFF. Earlier
  work by Ding\, Dub\\'edat\, Dunlap\, and Falconet (2019) showed that LFPP
  admits non-trivial subsequential limits. We show that the subsequential l
 imit is unique and satisfies our list of axioms. In the case when $\\gamma
  = \\sqrt{8/3}$\, our metric coincides with the $\\sqrt{8/3}$-LQG metric c
 onstructed in previous work by Miller and Sheffield\, which in turn is equ
 ivalent to the Brownian map for a certain variant of the GFF. For general 
 $\\gamma \\in (0\,2)$\, we conjecture that our metric is the Gromov-Hausdo
 rff limit of appropriate weighted random planar map models\, equipped with
  their graph distance.  \n\nBased on four joint papers with Jason Miller a
 nd one joint paper with Julien Dub\\'edat\, Hugo Falconet\, Josh Pfeffer\,
  and Xin Sun.\n
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
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