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SUMMARY:The fractal dimension of Liouville quantum gravity: monotonicity\,
  universality\, and bounds - Ewain Gwynne (Massachusetts Institute of Tech
 nology)
DTSTART:20180719T124500Z
DTEND:20180719T133000Z
UID:TALK108148@talks.cam.ac.uk
CONTACT:INI IT
DESCRIPTION:We show that for each $\\gamma \\in (0\,2)$\, there is an expo
 nent $d_\\gamma > 2$\, the ``fractal dimension of $\\gamma$-Liouville quan
 tum gravity (LQG)"\, which describes the ball volume growth exponent for c
 ertain random planar maps in the $\\gamma$-LQG universality class\, the gr
 aph-distance displacement exponent for random walk on these random planar 
 maps\, the exponent for the Liouville heat kernel\, and exponents for vari
 ous continuum approximations of $\\gamma$-LQG distances such as Liouville 
 graph distance and Liouville first passage percolation. This builds on wor
 k of Ding-Zeitouni-Zhang (2018).   We also show that $d_\\gamma$ is a cont
 inuous\, strictly increasing function of $\\gamma$ and prove upper and low
 er bounds for $d_\\gamma$ which in some cases greatly improve on previousl
 y known bounds for the aforementioned exponents. For example\, for $\\gamm
 a=\\sqrt 2$ (which corresponds to spanning-tree weighted planar maps) our 
 bounds give $3.4641 \\leq d_{\\sqrt 2} \\leq 3.63299$ and in the limiting 
 case we get $4.77485 \\leq \\lim_{\\gamma\\rightarrow 2^-} d_\\gamma \\leq
  4.89898$.   Based on joint works with Jian Ding\, Nina Holden\, Tom Hutch
 croft\, Jason Miller\, and Xin Sun.
LOCATION:Seminar Room 1\, Newton Institute
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