  The Erd\H{o}s--Ko--Rado (EKR) theorem and its generalizations can be viewed
as classifications of maximum independent sets in appropriately defined
families of graphs, such as the Kneser graph $K(n,k)$. In this paper, we
investigate the independence number of random spanning subraphs of two other
families of graphs whose maximum independent sets satisfy an EKR-type
characterization: the derangement graph on the set of permutations in
$\mathrm{Sym}(n)$ and the derangement graph on the set $\mathcal{M}_{n}$ of
perfect matchings in the complete graph $\mathcal{K}_{2n}$. In both cases, we
show there is a sharp threshold probability for the event that the independence
number of a random spanning subgraph is equal to that of the original graph. As
a useful tool to aid our computations, we obtain a Friedgut--Kalai--Naor (FKN)
type theorem on sparse boolean functions whose domain is the vertex set of
$\mathcal{M}_{n}$. In particular, we show that boolean functions whose Fourier
transforms are highly concentrated on the first two irreducible modules in the
$\mathrm{Sym}(2n)$ module $\mathbb{C}[\mathcal{M}_{n}]$, is close to being the
characteristic function of a union of maximum independent sets in the
derangement graph on perfect matchings.
