  Suitable cluster definitions have allowed researchers to describe many
ordering transitions in spin systems as geometric phenomena related to
percolation. For spin glasses and some other systems with quenched disorder,
however, such a connection has not been fully established, and the numerical
evidence remains incomplete. Here we use Monte Carlo simulations to study the
percolation properties of several classes of clusters occurring in the
Edwards-Anderson Ising spin-glass model in two dimensions. The
Fortuin-Kasteleyn-Coniglio-Klein clusters originally defined for the
ferromagnetic problem do percolate at a temperature that remains non-zero in
the thermodynamic limit. On the Nishimori line, this location is accurately
predicted by an argument due to Yamaguchi. More relevant for the spin-glass
transition are clusters defined on the basis of the overlap of several
replicas. We show that various such cluster types have percolation thresholds
that shift to lower temperature by increasing the system size, in agreement
with the zero-temperature spin-glass transition in two dimensions. The overlap
is linked to the difference in density of the two largest clusters, thus
supporting a picture where the spin-glass transition corresponds to an emergent
density difference of the two largest clusters inside the percolating phase.
