  The properties of zero modes in particle-hole symmetric systems are analyzed
in the presence of strong random scattering by a disordered environment. The
study is based on the calculation of the time-averaged density distribution on
a lattice. In particular, a flat distribution is found for strong random
scattering. This result is compared with a decaying distribution for weak
random scattering by an analysis of the scattering paths. In the calculation we
consider the invariant measure of the average two-particle Green's function,
which is related to lattice-covering self-avoiding (LCSA) strings. In
particular, strong scattering is associated with LCSA loops, whereas weaker
scattering is associated with open LCSA strings. Our results are a
generalization of the delocalized state observed at the band center of a
one-dimensional tight-binding model with random hopping by Dyson in 1953.
