  We study a model of an i.i.d.~random environment in general dimensions $d\ge
2$, where each site is equipped with one of two environments. The model comes
with a parameter $p$ which governs the frequency of the first environment, and
for each dimension $d$ there is a critical parameter $p_c(d)$ at which there is
a phase transition for the geometry of a particular connected cluster (the
cluster is infinite for all $p$). We use the celebrated methodology of
enhancements in this novel setting to prove that $p_c(d)$ is strictly monotone
in $d$ for this model. To do so we study the discrete geometry and percolation
theory of higher-dimensional structures called terraces.
