  We study the density of clustered immobile reactants in the
diffusion-controlled single species annihilation. An initial state in which
these impurities occupy a subspace of codimension d' leads to a substantial
enhancement of their survival probability. The Smoluchowski rate theory
suggests that the codimensionality plays a crucial role in determining the long
time behavior. The system undergoes a transition at d'=2. For d'<2, a finite
fraction of the impurities survive: ni(t) ~ ni(infinity)+const x log(t)/t^{1/2}
for d=2 and ni(t) ~ ni(infinity)+const/t^{1/2} for d>2. Above this critical
codimension, d'>=2, the subspace decays indefinitely. At the critical
codimension, inverse logarithmic decay occurs, ni(t) ~ log(t)^{-a(d,d')}. Above
the critical codimension, the decay is algebraic ni(t) ~ t^{-a(d,d')}. In
general, the exponents governing the long time behavior depend on the dimension
as well as the codimension.

Condensed Matter