Simulations of random planar maps and discrete Liouville quantum gravity contradict Watabiki's formula for the Hausdorff dimension and support the Ding-Gwynne formula for central charges in [-12.5, 0).
Heat kernel for Liouville Brownian motion and Liouville graph distance
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abstract
We show the existence of the scaling exponent $\chi\in (0,4[(1+\gamma^2/4)- \sqrt{1+\gamma^4/16}]/\gamma^2]$ of the graph distance associated with subcritical two-dimensional Liouville quantum gravity of paramater $\gamma<2$ on $\mathbb V =[0,1]^2 $. We also show that the Liouville heat kernel satisfies, for any fixed $u,v\in \mathbb V^o$, the short time estimates $$ \lim_{ t \to 0} \frac{\log |\log {\mathsf p}_t^\gamma(u,v)| }{|\log t|}=\frac{\chi}{2-\chi}, \ \mbox{\rm a.s.} $$
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gr-qc 1years
2019 1verdicts
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Precision measurements of Hausdorff dimensions in two-dimensional quantum gravity
Simulations of random planar maps and discrete Liouville quantum gravity contradict Watabiki's formula for the Hausdorff dimension and support the Ding-Gwynne formula for central charges in [-12.5, 0).