Scaling limit of the collision measure for two-dimensional random walks
Abstract
We study the scaling limit of the collision measure of two i.i.d. discrete-time random walks on ℤ2, which records their collision sites and times. This is a critical regime for collisions: the walks collide infinitely often, whereas the limiting Brownian motions do not collide at positive times. Hence, existing general results on the convergence of collision measures do not apply. Assuming that the jump distribution has mean zero and finite second moment, we prove that, under a logarithmic scaling, the collision measure converges in distribution to a non-trivial random measure. We also give an explicit representation of the limiting measure in terms of a Poisson point process. To prove convergence, we introduce a new cluster decomposition of the Laplace transform of the collision measure, revealing the cluster structure of collisions.
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BibTeX
@misc{nakajima2026scaling,
title = {{Scaling limit of the collision measure for two-dimensional random walks}},
author = {Shuta Nakajima and Ryoichiro Noda},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03115},
eprint = {2610.03115},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03115},
}
RIS
TY - UNPB
TI - Scaling limit of the collision measure for two-dimensional random walks
AU - Shuta Nakajima
AU - Ryoichiro Noda
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03115
UR - https://arxiv.org/abs/2610.03115
AB - We study the scaling limit of the collision measure of two i.i.d. discrete-time random walks on ℤ^2, which records their collision sites and times. This is a critical regime for collisions: the walks collide infinitely often, whereas the limiting Brownian motions do not collide at positive times. Hence, existing general results on the convergence of collision measures do not apply. Assuming that the jump distribution has mean zero and finite second moment, we prove that, under a logarithmic scaling, the collision measure converges in distribution to a non-trivial random measure. We also give an explicit representation of the limiting measure in terms of a Poisson point process. To prove convergence, we introduce a new cluster decomposition of the Laplace transform of the collision measure, revealing the cluster structure of collisions.
ER -