  In this work, we consider a two-dimensional square lattice of pinned magnetic
spins with nearest-neighbour interactions and we randomly replace a fixed
proportion of spins with nonmagnetic defects carrying no spin. We focus on the
linear spin-wave regime and address the propagation of a spin-wave excitation
with initial momentum $k_0$. We compute the disorder-averaged momentum
distribution obtained at time $t$ and show that the system exhibits two
regimes. At low defect density, typical disorder configurations only involve a
single percolating magnetic cluster interspersed with single defects
essentially and the physics is driven by Anderson localization. In this case,
the momentum distribution features the emergence of two known emblematic
signatures of coherent transport, namely the coherent backscattering (CBS) peak
located at $-k_0$ and the coherent forward scattering (CFS) peak located at
$k_0$. At long times, the momentum distribution becomes stationary. However,
when increasing the defect density, site percolation starts to set in and
typical disorder configurations display more and more disconnected clusters of
different sizes and shapes. At the same time, the CFS peak starts to oscillate
in time with well defined frequencies. These oscillation frequencies represent
eigenenergy differences in the regular, disorder-immune, part of the
Hamiltonian spectrum. This regular spectrum originates from the small-size
magnetic clusters and its weight grows as the system undergoes site percolation
and small clusters proliferate. Our system offers a unique spectroscopic
signature of cluster formation in site percolation problems.
