  We develop a theory of amorphous interfaces in glass-forming liquids. We show
that the statistical properties of these surfaces, which separate regions
characterized by different amorphous arrangements of particles, coincide with
the ones of domain walls in the random field Ising model. A major consequence
of our results is that super-cooled liquids are characterized by two different
static lengths: the point-to-set $\xi_{PS}$ which is a measure of the spatial
extent of cooperative rearranging regions and the wandering length $\xi_\perp$
which is related to the fluctuations of their shape. We find that $\xi_\perp$
grows when approaching the glass transition but slower than $\xi_{PS}$. The
wandering length increases as $s_c^{-1/2}$, where $s_c$ is the configurational
entropy. Our results strengthen the relationship with the random field Ising
model found in recent works. They are in agreement with previous numerical
studies of amorphous interfaces and provide a theoretical framework for
explaining numerical and experimental findings on pinned particle systems and
static lengths in glass-forming liquids.
