  We study the properties of the Nakajima-Zwanzig memory kernel for a qubit
immersed in a many-body localized (i.e., disordered and interacting) bath. We
argue that the memory kernel decays as a power law in both the localized and
ergodic regimes, and show how this can be leveraged to extract $t\to\infty$
populations for the qubit from finite time ($J t \leq 10^2$) data in the
thermalizing phase. This allows us to quantify how the long-time values of the
populations approach the expected thermalized state as the bath approaches the
thermodynamic limit. This approach should provide a good complement to
state-of-the-art numerical methods, for which the long-time dynamics with large
baths are impossible to simulate in this phase. Additionally, our numerics on
finite baths reveal the possibility for unbounded exponential growth in the
memory kernel, a phenomenon rooted in the appearance of exceptional points in
the projected Liouvillian governing the reduced dynamics. In small systems
amenable to exact numerics, we find that these pathologies may have some
correlation with delocalization.
