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The fractal dimension of Liouville quantum gravity: monotonicity, universality, and bounds

Presented by: 
Ewain Gwynne Massachusetts Institute of Technology
Date: 
Thursday 19th July 2018 - 13:45 to 14:30
Venue: 
INI Seminar Room 1
Abstract: 
We show that for each $\gamma \in (0,2)$, there is an exponent $d_\gamma > 2$, the ``fractal dimension of $\gamma$-Liouville quantum gravity (LQG)", which describes the ball volume growth exponent for certain random planar maps in the $\gamma$-LQG universality class, the graph-distance displacement exponent for random walk on these random planar maps, the exponent for the Liouville heat kernel, and exponents for various continuum approximations of $\gamma$-LQG distances such as Liouville graph distance and Liouville first passage percolation. This builds on work of Ding-Zeitouni-Zhang (2018). We also show that $d_\gamma$ is a continuous, strictly increasing function of $\gamma$ and prove upper and lower bounds for $d_\gamma$ which in some cases greatly improve on previously known bounds for the aforementioned exponents. For example, for $\gamma=\sqrt 2$ (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 Hutchcroft, Jason Miller, and Xin Sun.
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University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons