  We investigate a rare instance of an exactly solvable nonequilibrium
many-body problem. In particular, we derive an exact solution for the
nonequilibrium dynamics of an initially localized single hole in a fully
anisotropic antiferromagnetic Bethe lattice, described by the $t$-$J_z$ model.
The solvability of the model relies on the fractal self-similarity of Bethe
lattices, making it possible to compute the full motion of the hole as it moves
through the lattice, as well as exactly characterizing the resulting effect on
spin-spin correlation functions. We find that the hole remains bound to its
initial position with large aperiodic oscillations in the hole density
distribution. We track this back to the irregular pattern of the eigenenergies
of the magnetic polaron ground state and string excitations, which we also
determine exactly.
