  This contribution is not intended as a review but, by suggestion of the
editors, as a glimpse ahead into the realm of dually weighted tensor models for
quantum gravity. This class of models allows one to consider a wider class of
quantum gravity models, in particular one can formulate state sum models of
spacetime with an intrinsic notion of foliation. The simplest one of these
models is the one proposed by Benedetti and Henson, which is a matrix model
formulation of two-dimensional Causal Dynamical Triangulations (CDT). In this
paper we apply the Functional Renormalization Group Equation (FRGE) to the
Benedetti-Henson model with the purpose of investigating the possible continuum
limits of this class of models. Possible continuum limits appear in this FRGE
approach as fixed points of the renormalization group flow where the size of
the matrix acts as the renormalization scale. Considering very small
truncations, we find fixed points that are compatible with analytically known
results for CDT in two dimensions. By studying the scheme dependence of our
results we find that precision results require larger truncations than the ones
considered in the present work. We conclude that our work suggests that the
FRGE is a useful exploratory tool for dually weighted matrix models. We thus
expect that the FRGE will be a useful exploratory tool for the investigation of
dually weighted tensor models for CDT in higher dimensions.
