Scaling limit of the collision measures of multiple random walks

Abstract

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.

Burstiness and information spreading in active particle systems

“…From a theoretical perspective, we realize that this unusual result that B is large at large noise strengths should connect to the meeting or returning problems of random walks. [46][47][48] The periodic boundary conditions and the finite size effect can also affect the meeting rate of two particles. Meanwhile, the powerlaw exponent a E 1.3 we found for the interevent time distribution should also relate to the meeting problems of the random walks.…”