  We obtain large deviations estimates for the self-intersection local times
for a symmetric random walk in dimension 3. Also, we show that the main
contribution to making the self-intersection large, in a time period of length
$n$, comes from sites visited less than some power of $\log(n)$. This is
opposite to the situation in dimensions larger or equal to 5. Finally, we
present two applications of our estimates: (i) to moderate deviations estimates
for the range of a random walk, and (ii) to moderate deviations for random walk
in random sceneries.
