  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.
