Internal DLA is studied with a random, homogeneous, distribution of traps. Particles are injected at the
origin of a $d$-dimensional Euclidean lattice and perform independent random walks until they hit an unsaturated trap, at which
time the particle dies and the trap becomes saturated.
It is shown that the large scale effect of the
randomness of the traps on the speed of growth of the set of saturated traps depends of the strength of the injection, and
separates into several regimes. In the subcritical regime, the set of saturated traps is asymptotically a Euclidean ball whose
radius is determined in a trivial way from the trap density.
In the critical regime, there is a nontrivial interplay between the density of traps and the rate of growth of
the ball. The supercritical regime is studied using order statistics for
free random walks. This restricts us to $d=1$.
In the supercritical, subexponential regime, there is an overall effect of the
traps, but their density does not affect the growth rate. Finally, in the supercritical,
superexponential regime, the traps have no effect at all, and the asymptotics is governed by that
of free random walks on the lattice.