arXiv:2511.11351v2 Announce Type: replace Abstract: Motivated by recent high-dimensional reliability algorithms, we study a random matrix model corresponding to the estimation by importance sampling of $\Sigma_A$, the covariance matrix of a Gaussian vector conditioned by a rare event $A$. For such models, we show that, in contrast to most random matrix models, the minimal sample size to ensure an accurate estimation of $\Sigma_A$ needs to grow superlinearly in the dimension. More precisely, if $n$ is the sample size and $d$ the dimension, we show that the estimation of $\Sigma_A$ is accurate if $n/d \gg L$ and not accurate if $n/d \ll L$ where $L$ is the maximum likelihood ratio. Moreover, we relate the asymptotic behavior of $L$ to $\lambda_1$, the smallest eigenvalue of the auxiliary covariance matrix, which suggests that $L$ (and thus the minimal sample size) will be larger for small values of $\lambda_1$. Numerical experiments confirm that this insight carries over to recent cross-entropy algorithms, with the best performing algorithms being also the ones with the largest smallest eigenvalue. Although backed up by our theoretical result, this result is surprising as these algorithms were not designed to act on the spectrum.