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PreprintCC BY 4.0arXiv

Point Configurations in Brownian Traces

  • P. Kosenko
  • J. M. Medina
  • A. Yavicoli
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Abstract

We study finite and countable point configurations in the trace of Brownian motion in ℝd, d≥2. In the plane, the set of translations placing a prescribed finite configuration on the trace has Hausdorff dimension two in every nonempty open set; we prove analogous dimension-preservation results for radial and bilipschitz patterns, and every prescribed countable configuration has a dense translation set. For d≥3 we prove a sharp occurrence criterion for regular scale-invariant families of configurations, in terms of the ranks of all labelled subconfigurations. It gives exact thresholds for arithmetic progressions and for homothetic and similar copies of finite sets. In particular we recover and extend a result of Benjamini and Kozma: almost surely the longest arithmetic progression in the trace has length five in ℝ3, three in ℝ4 and ℝ5, and two in ℝd for d≥6.

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Cite

BibTeXRIS
BibTeX
@misc{kosenko2026point,
  title = {{Point Configurations in Brownian Traces}},
  author = {P. Kosenko and J. M. Medina and A. Yavicoli},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04000},
  eprint = {2610.04000},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04000},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Point Configurations in Brownian Traces
AU  - P. Kosenko
AU  - J. M. Medina
AU  - A. Yavicoli
PY  - 2026
DA  - 2026/10/02
PB  - arXiv
DO  - 10.48550/arxiv.2610.04000
UR  - https://arxiv.org/abs/2610.04000
AB  - We study finite and countable point configurations in the trace of Brownian motion in ℝ^d, d≥2. In the plane, the set of translations placing a prescribed finite configuration on the trace has Hausdorff dimension two in every nonempty open set; we prove analogous dimension-preservation results for radial and bilipschitz patterns, and every prescribed countable configuration has a dense translation set. For d≥3 we prove a sharp occurrence criterion for regular scale-invariant families of configurations, in terms of the ranks of all labelled subconfigurations. It gives exact thresholds for arithmetic progressions and for homothetic and similar copies of finite sets. In particular we recover and extend a result of Benjamini and Kozma: almost surely the longest arithmetic progression in the trace has length five in ℝ^3, three in ℝ^4 and ℝ^5, and two in ℝ^d for d≥6.
ER  -