Percolation of terraces, and enhancements for the orthant model

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.