We consider the XX and Heisenberg chains in a random field, mapping to free or interacting fermions in a random potential. For finite sizes, the magnetization can never be exactly 1/2, but we focus on the place in the chain where it comes extremely close. In the top panel we show ED results for the deviation from this perfect polarization in a random (high energy) eigenstate in the XX chain (Sz=0 sector). Some sites are almost perfectly polarized. With increasing size they become more polarized, leading to a chain break. This corresponds to a half-filled, "many-body" Anderson chain with a random on-site potential. All the single particle orbitals are exponentially localized. Zooming in on the most polarized site, we see that it corresponds to occupying a number of neighboring such orbitals. As the bottom left panel illustrates, a simple toy model of exponentially localized occupied or empty orbitals, all with the same localization length, can already capture this simple mechanism that leads to a chain breaking. (It can also predict the chain breaking exponent! ) Furthermore, we find that the distribution of the minimum deviations (maximum polarizations) can be described by Extreme Value Theory (EVT), as shown by the fits in the upper right panel. Between the toy model and EVT, the main properties of these distributions can be understood. Finally, considering minimal deviations of the Heisenberg chain (the interacting many-body problem!), we are limited to much smaller sizes, and simple EVT does not exactly apply. But we can compare with the XX chain distributions using the Kullback-Leibler divergence! With this, we observe an "extreme-statistics transition" as a function of the disorder strength: at strong disorder, the distributions are close to those of the localized, many-body Anderson chain, but at weak disorder, they are extremely different. Now for the killer question - does this transition correspond to the MBL transition ? 🤔 (Spoiler: more work is needed! 😉) Questions and comments welcome!