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This works investigates the Lyapunov-Oseledets spectrum of transfer operatorcocycles associated to one-dimensional random paired tent maps depending on aparameter $\epsilon$, quantifying the strength of the \emph{leakage} betweentwo nearly invariant regions. We show that the system exhibits metastability,and identify the second Lyapunov exponent $\lambda_2^\epsilon$ within an errorof order $\epsilon^2|\log \epsilon|$. This approximation agrees with the naiveprediction provided by a time-dependent two-state Markov chain. Furthermore, itis shown that $\lambda_1^\epsilon=0$ and $\lambda_2^\epsilon$ are simple, andare the only exceptional Lyapunov exponents of magnitude greater than $-\log2+O(\log\log\tfrac 1\epsilon/\log\tfrac 1\epsilon)$.

Lyapunov exponents for transfer operator cocycles of metastable maps: a quarantine approach