  We numerically investigate the magnon-mediated spin transport and scaling
behaviors of a nonmagnetic metal/nonmagnetic metal/ferromagnetic insulator
(NM/NM/FI) heterostructure in the presence of Anderson disorders. For the
two-dimensional (2D) NM/NM/FI system, an enhancement of spin conductance in the
weak disorder regime is found due to the increasing of the interfacial density
of states (DOS) at NM/FI interface. As a result, a new scaling regime is
uncovered in the metallic regime where the spin conductance fluctuation rms(G)
scales linearly with the average spin conductance <G>, independent of system
parameters such as Fermi energies and temperatures. The competition between the
disorder-enhanced interfacial DOS and disorder-suppressed spin transport
results in a non-monotonic dependence of average spin conductance on disorder
strength. In the localized regime, the variance of lnG for different system
parameters follows a universal function that depends linearly on the average of
logarithm of spin conductance <lnG>.The distribution of lnG in the localized
regime is non-Gaussian whose deviation from Gaussian can be characterized by
the third and fourth order cumulants of lnG, denoted as k3 and k4,
respectively. We find that k3~<-lnG>^3/2 and k4~<lnG>^2. This suggests that the
spin conductance mediated by magnon in NM/NM/FI hybrid systems belongs to a
different universality class. For fixed <lnG>, the lnG distribution in the
localized regime for different system parameters is found to collapse onto a
single curve, suggesting the distribution P(lnG;<lnG>) is a universal function
that depends only on <lnG>. Furthermore, spin thermopower S in the presence of
disorder is studied and the variance of lnS is found to scale linearly with
<lnS> for different system parameters in the localized regime. Spin conductance
and spin thermopower for 1D NM/NM/FI are also studied.
