  Although spin foams arose as quantizations of the length metric degrees of
freedom, the quantum configuration space is rather based on areas as more
fundamental variables. This is also highlighted by the semi-classical limit of
four-dimensional spin foam models, which is described by the Area Regge action.
Despite its central importance to spin foams the dynamics encoded by the Area
Regge action is only poorly understood, in particular in the continuum limit.
We perform here a systematic investigation of the dynamics defined by the Area
Regge action on a regular centrally subdivided hypercubical lattice. This
choice of lattice avoids many problems of the non-subdivided hypercubical
lattice, for which the Area Regge action is singular. The regularity of the
lattice allows to extract the continuum limit and its corrections, order by
order in the lattice constant. We show that, contrary to widespread
expectations which arose from the so-called flatness problem of spin foams, the
continuum limit of the Area Regge action does describe to leading order the
same graviton dynamics as general relativity. The next-to-leading order
correction to the effective action for the length metric is of second order in
the lattice constant, and is given by a quadratic term in the Weyl curvature
tensor. This correction can be understood to originate from an underlying
dynamics of area metrics. This suggests that the continuum limit of spin foam
dynamics does lead to massless gravitons, and that the leading order quantum
corrections can be understood to emerge from a generalization of the
configuration space from length to area metrics.
