For an integer k ≥ 2, let S (1) , S (2) , . . . , S (k) be k independent simple symmetric random walks on Z. A pair (n, z) is called a collision event if there are at least two distinct random walks, namely, S (i) , S (j) satisfyingWe 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] × 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.

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