  We consider the problem of determining the spectrum of an electronic spin via
polarization transfer to coupled nuclear spins and their subsequent readout.
This suggests applications for employing dynamic nuclear polarization (DNP) for
"ESR-via-NMR". In this paper, we describe the theoretical basis for this
process by developing a model for the evolution dynamics of the coupled
electron-nuclear system through a cascade of Landau-Zener anti-crossings
(LZ-LACs). We develop a method to map these traversals to the operation of an
equivalent "Galton board". Here, LZ-LAC points serve as analogues to Galton
board "pegs", upon interacting with which the nuclear populations redistribute.
The developed hyperpolarization then tracks the local electronic density of
states. We show that this approach yields an intuitive and analytically
tractable solution of the polarization transfer dynamics, including when DNP is
carried out at the wing of a homogeneously broadened electronic spectral line.
We apply this approach to a model system comprised of a Nitrogen Vacancy (NV)
center electron in diamond, hyperfine coupled to N neighboring 13C nuclear
spins, and discuss applications for nuclear-spin interrogated NV center
magnetometry. More broadly, the methodology of "one-to-many"
electron-to-nuclear spectral mapping developed here suggests interesting
applications in quantum memories and sensing, as well as wider applications in
modeling DNP processes in the multiple nuclear spin limit.
