  A number of approaches to 4D quantum gravity, such as holography and loop
quantum gravity, propose areas instead of lengths as fundamental variables. The
Area Regge action, which can be defined for general 4D triangulations, is a
natural choice for an action based on areas. It does indeed appear in the
semi-classical limit of spin foam models. The Area Regge action does however
only lead to a discrete version of the gravitational equations of motion, if
one implements constraints, that ensure that the areas are compatible with a
consistent length assignment to the edges of the triangulation. The constrained
version is then classically equivalent to the Length Regge action, which
provides a discretization of the Einstein-Hilbert action.
  Here we perform the first systematic analysis of the Area Regge dynamics on a
hyper-cubical lattice. Surprisingly, we find that the linearized Area Regge
action on a hyper-cubical lattice does single out the Length Regge action by
its scaling behaviour in the lattice constant. That is, integrating out the
variables describing fluctuations in the area-length constraints one finds the
linearized Length Regge action plus terms of higher order in the lattice
constant. This appears without any explicit implementation of the area-length
constraints.
