  The Dvali-Gabadadze-Porrati model introduces a parameter, the cross-over
scale $r_c$, setting the scale where higher dimensional effects are important.
In order to agree with observations and to explain the current acceleration of
the Universe, $r_c$ must be of the order of the present Hubble radius. We
discuss a mechanism to generate a large $r_c$, assuming that it is determined
by a dynamical field and exploiting the quantum effects of the graviton. For
simplicity, we consider a scalar field $\Psi$ with a kinetic term on the brane
instead of the full metric perturbations. We compute the Green function and the
1-loop expectation value of the stress tensor of $\Psi$ on the background
defined by a flat bulk and an inflating brane (self-accelerated or not). We
also include the flat brane limit. The quantum fluctuations of the bulk field
$\Psi$ provide an effective potential for $r_c$. For a flat brane, the 1-loop
effective potential is of the Coleman-Weinberg form, and admits a minimum for
large $r_c$ without fine tuning. When we take into account the brane curvature,
a sizeable contribution at the classical level changes this picture and the
potential develops a (minimum) maximum for the (non-) self-accelerated branch.
