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

## TL;DR

This paper introduces a discretized model of Liouville quantum gravity (LQG) valid for all matter central charges in (-infinity, 25), extending formulas and exploring geometric properties in the supercritical regime.

## Contribution

It develops a new discretization of LQG that remains meaningful for matter central charge greater than 1, extending the KPZ formula and analyzing geometric dimensions.

## Key findings

- Formulas for dimension-related quantities hold for c_M in (1,25)
- Graph distances grow polynomially, but ball sizes grow faster than any power
- Hausdorff dimension of LQG appears infinite for c_M in (1,25)

## 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.