  For an integer $k\ge 2$, let $S^{(1)}, S^{(2)}, \dots, S^{(k)}$ be $k$
independent simple symmetric random walks on $\mathbb{Z}$. A pair $(n,z)$ is
called a collision event if there are at least two distinct random walks,
namely, $S^{(i)},S^{(j)}$ satisfying $S^{(i)}_n= S^{(j)}_n=z$. We show that
under the same scaling as in Donsker's theorem, the sequence of random measures
representing these collision events converges to a non-trivial random measure
on $[0,1]\times \mathbb{R}$. Moreover, the limit random measure can be
characterized using Wiener chaos. The proof is inspired by methods from
statistical mechanics, especially, by a partition function that has been
developed for the study of directed polymers in random environments.
