  We consider quasi-compact linear operator cocycles
$\mathcal{L}^{n}_\omega:=\mathcal{L}_{\sigma^{n-1}\omega}\circ\cdots\circ\mathcal{L}_{\sigma\omega}\circ
\mathcal{L}_{\omega}$ driven by an invertible ergodic process
$\sigma:\Omega\to\Omega$, and their small perturbations
$\mathcal{L}_{\omega,\epsilon}^{n}$. We prove an abstract $\omega$-wise
first-order formula for the leading Lyapunov multipliers. We then consider the
situation where $\mathcal{L}_{\omega}^{n}$ is a transfer operator cocycle for a
random map cocycle $T_\omega^{n}:=T_{\sigma^{n-1}\omega}\circ\cdots\circ
T_{\sigma\omega}\circ T_{\omega}$ and the perturbed transfer operators
$\mathcal{L}_{\omega,\epsilon}$ are defined by the introduction of small random
holes $H_{\omega,\epsilon}$ in $[0,1]$, creating a random open dynamical
system. We obtain a first-order perturbation formula in this setting, which
reads
$\lambda_{\omega,\epsilon}=\lambda_\omega-\theta_{\omega}\mu_{\omega}(H_{\omega,\epsilon})+o(\mu_\omega(H_{\omega,\epsilon})),$
where $\mu_\omega$ is the unique equivariant random measure (and equilibrium
state) for the original closed random dynamics. Our new machinery is then
deployed to create a spectral approach for a quenched extreme value theory that
considers random dynamics with general ergodic invertible driving, and random
observations. An extreme value law is derived using the first-order terms
$\theta_\omega$. Further, in the setting of random piecewise expanding interval
maps, we establish the existence of random equilibrium states and conditionally
invariant measures for random open systems via a random perturbative approach.
Finally we prove quenched statistical limit theorems for random equilibrium
states arising from contracting potentials. We illustrate the theory with a
variety of explicit examples.
