W0_sample = np.random.normal(0,1)?

Abstract: In this note we explore the distribution of vacuum expectation values of the superpotential $W_0$ in explicit Type IIB flux compactifications. We show that the distribution can be approximated, universally across geometries, by a two-dimensional Gaussian with a model dependent standard deviation. We identify this behaviour in 20 Calabi-Yau orientifold compactifications with between two and five complex structure moduli by constructing a total of $\mathcal{O}(10^7)$ flux vacua. We observe a characteristic scaling behaviour of the width $\sigma$ of our distributions with respect to the D3-charge contributions $N_{\text{flux}}$ from fluxes which can be approximated by $\sigma \sim \sqrt{N_{\text{flux}}}$. This $W_0$ distribution implies that locating small values of $|W_0|$ as a preferred regime associated with classes of string theory solutions typically featuring hierarchies, simplifies to the basic statement of finding small Euclidean norms of normally distributed values. We do also identify small modifications to this Gaussian behaviour in our samples which might be seen as indications for the breakdown of the continuous flux approximation commonly used in the context of statistical analyses of the flux landscape.