Liouville quantum gravity with matter central charge in $(1,25)$: a probabilistic approach

Abstract

There is a substantial literature concerning Liouville quantum gravity (LQG) in two dimensions with conformal matter field of central charge ${\mathbf{c}}_{\mathrm{M}}\in (-\infty ,1]$. Via the DDK ansatz, LQG can equivalently be described as the random geometry obtained by exponentiating $\gamma$ times a variant of the planar Gaussian free field (GFF), where $\gamma \in (0,2]$ satisfies ${\mathbf{c}}_{\mathrm{M}}=25-6(2/\gamma +\gamma /2{)}^2$. Physics considerations suggest that LQG should also make sense in the regime when ${\mathbf{c}}_{\mathrm{M}}>1$. However, the behavior in this regime is rather mysterious in part because the corresponding value of $\gamma$ is complex, so analytic continuations of various formulas give complex answers which are difficult to interpret in a probabilistic setting. We introduce and study a discretization of LQG which makes sense for all values of ${\mathbf{c}}_{\mathrm{M}}\in (-\infty ,25)$. Our discretization consists of a random planar map, defined as the adjacency graph of a tiling of the plane by dyadic squares which all have approximately the same "LQG size" with respect to the GFF. We prove that several formulas for dimension-related quantities are still valid for ${\mathbf{c}}_{\mathrm{M}}\in (1,25)$, with the caveat that the dimension is infinite when the formulas give a complex answer. In particular, we prove an extension of the (geometric) KPZ formula for ${\mathbf{c}}_{\mathrm{M}}\in (1,25)$, which gives a finite quantum dimension iff the Euclidean dimension is at most $(25-{\mathbf{c}}_{\mathrm{M}})/12$. We also show that the graph distance between typical points with respect to our discrete model grows polynomially whereas the cardinality of a graph distance ball of radius $r$ grows faster than any power of $r$ (which suggests that the Hausdorff dimension of LQG is infinite for ${\mathbf{c}}_{\mathrm{M}}\in (1,25)$). We include a substantial list of open problems.

arXiv comment53 pages, 6 figures

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• Weak Liouville quantum gravity metrics with matter central charge $\mathbf{c}\in (-\infty ,25)$
• Punctures and p-spin curves from matrix models III. Dl type and logarithmic potential
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• Conformally invariant random fields, quantum Liouville measures, and random Paneitz operators on Riemannian manifolds of even dimension