  For any $d \geq 1$, random $\mathbb{Z}^d$ shifts of finite type (SFTs) were
defined in previous work of the authors. For a parameter $\alpha \in [0,1]$, an
alphabet $\mathcal{A}$, and a scale $n \in \mathbb{N}$, one obtains a
distribution of random $\mathbb{Z}^d$ SFTs by randomly and independently
forbidding each pattern of shape $\{1,\dots,n\}^d$ with probability $1-\alpha$
from the full shift on $\mathcal{A}$. We prove two main results concerning
random $\mathbb{Z}^d$ SFTs. First, we establish sufficient conditions on
$\alpha$, $\mathcal{A}$, and a $\mathbb{Z}^d$ subshift $Y$ so that a random
$\mathbb{Z}^d$ SFT factors onto $Y$ with probability tending to one as $n$
tends to infinity. Second, we provide sufficient conditions on $\alpha$,
$\mathcal{A}$ and a $\mathbb{Z}^d$ subshift $X$ so that $X$ embeds into a
random $\mathbb{Z}^d$ SFT with probability tending to one as $n$ tends to
infinity.
