  We study the relation between the hessian matrix of the riemannian Reggae
action on a 4-simplex and linearized quantum gravity. We give an explicit
formula for the hessian as a function of the geometry, and show that it has a
single zero mode. We then use a 3d lattice model to show that (i) the zero mode
is a remnant of the continuum diffeomorphism invariance, and (ii) we recover
the complete free graviton propagator in the continuum limit. The results help
clarify the structure of the boundary state needed in the recent calculations
of the graviton propagator in loop quantum gravity, and in particular its role
in fixing the gauge.
