  The hyperfine coupling between the spin of a charge carrier and the nuclear
spin bath is a predominant channel for the carrier spin relaxation in many
organic semiconductors. We theoretically investigate the hyperfine-induced spin
relaxation of a carrier performing a random walk on a d-dimensional regular
lattice, in a transport regime typical for organic semiconductors. We show that
in d=1 and d=2 the time dependence of the space-integrated spin polarization,
P(t), is dominated by a superexponential decay, crossing over to a stretched
exponential tail at long times. The faster decay is attributed to multiple
self-intersections (returns) of the random walk trajectories, which occur more
often in lower dimensions. We also show, analytically and numerically, that the
returns lead to sensitivity of P(t) to external electric and magnetic fields,
and this sensitivity strongly depends on dimensionality of the system (d=1 vs.
d=3). Furthermore, we investigate in detail the coordinate dependence of the
time-integrated spin polarization, $\sigma(r)$, which can be probed in the spin
transport experiments with spin-polarized electrodes. We demonstrate that,
while $\sigma(r)$ is essentially exponential, the effect of multiple
self-intersections can be identified in transport measurements from the strong
dependence of the spin decay length on the external magnetic and electric
fields.
