  We consider the annealed asymptotics for the survival probability of Brownian
motion among randomly distributed traps. The configuration of the traps is
given by independent displacements of the lattice points. We determine the long
time asymptotics of the logarithm of the survival probability up to a
multiplicative constant. As applications, we show the Lifshitz tail effect of
the density of states of the associated random Schr\"{o}dinger operator and
derive a quantitative estimate for the strength of intermittency in the
Parabolic Anderson problem.
