  We consider the Lattice Gaussian free field in $d+1$ dimensions, $d=3$ or
larger, on a large box (linear size $N$) with boundary conditions zero. On this
field two potentials are acting: one, that models the presence of a wall,
penalizes the field when it enters the lower half space and one, the
{\guillemotleft}pinning potential{\guillemotright} , that rewards visits to the
proximity of the wall. The wall can be soft, i.e. the field has a finite
penalty to enter the lower half plane, or hard when the penalty is infinite. In
general the pinning potential is disordered and it gives on average a reward h
in $\mathbb{R}$ (a negative reward is a penalty): the energetic contribution
when the field at site x visits the pinning region is $\beta \omega_x+h$,
$\{\omega_x\}_{x \in \mathbb{Z}^d}$ are IID centered and exponentially
integrable random variables of unit variance and $\beta\ge 0$. In [E.
Bolthausen, J.-D. Deuschel and O. Zeitouni, J. Math. Phys. 41 (2000),
1211-1223] it is shown that, when $\beta=0$ (that is, in the non disordered
model), a delocalization-localization transition happens at $h=0$, in
particular the free energy of the system is zero for $h \le 0$ and positive for
$h>0$. We show that, for $\beta\neq 0$, the transition happens at
$h=h_c(\beta):=- \log \mathbb{E} \exp(\beta \omega_x)$ and we find the precise
asymptotic behavior of the logarithm of the free energy density of the system
when $h \searrow h_c(\beta)$. In particular, we show that the transition is of
infinite order in the sense that the free energy is smaller than any power of
$h-h_c(\beta)$ in the neighborhood of the critical point and that disorder does
not modify at all the nature of the transition. We also provide results on the
behavior of the paths of the random field in the limit $N \to \infty$.
