  We numerically study the real-time dynamics of a single hole created in the
$t-J$ model on a square lattice. Initially, the hole spreads ballistically with
a velocity proportional to the hopping matrix element. At intermediate to long
times, the dimensionality as well as the spin background determine the hole
dynamics. A hole created in the ground state of a two dimensional quantum
antiferromagnet propagates again ballistically at long times but with a
velocity proportional to the spin exchange coupling, showing the formation of a
magnetic polaron. We provide an intuitive explanation of this dynamics in terms
of a parton construction, which leads to a good quantitative agreement with the
numerical simulations. In the limit of infinite temperature and no spin
exchange couplings, the dynamics can be approximated by a quantum random walk
on the Bethe lattice. Adding Ising interactions corresponds to an effective
disordered potential, which can dramatically slow down the hole propagation,
consistent with subdiffusive dynamics.
