Skip to content
DraconiticResearch
Search Sources API
190,335 studies31 sourcesUpdated 22 min ago
PreprintCC BY 4.0arXiv

Realising atomless laws as distance distributions on metric measure spaces

  • Christoph Thäle
  • Philipp Tuchel
Open at sourcePDFDOICite

Abstract

We study which probability measures on [0,∞) can occur as the distribution of the distance between two independent points sampled from a complete separable metric space equipped with a Borel probability measure. We prove that an atomless Borel probability measure can be realised in this way if and only if its support contains the origin. This settles the conjecture of Aldous, Blanc, and Curien for absolutely continuous laws and also covers atomless measures that are singular with respect to Lebesgue measure. Moreover, for every L>1, we show that the realising metric may be chosen L-bi-Lipschitz equivalent to an ultrametric. The proof constructs the space from compact components whose mutual distances produce prescribed parts of the measure, while distances within the components are assigned to smaller scales.

Related

  1. Connes spectral distance as a quantum distance measure and a geometric measure of entanglement
    Bing-Sheng Lin, Ji-Hong Wang
    arXivPreprintCC BY 4.0Full textMathematical physicsarXiv
  2. The Sunada method on Metric Measure Spaces
    Lewis Tadman
    arXivPreprintarXiv licenceFull textAnalysis & PDEsGeometry & topologyarXiv
  3. Total Variation Distance between Product Distributions: an Analytic Proxy
    Ariel Avital, Aryeh Kontorovich, Roman Vershynin +1
    arXivPreprintarXiv licenceFull textProbabilityStatisticsarXiv
  4. Coarse geometry of metric measure spaces
    Takayuki Okuda, Takashi Shioya
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv
  5. Lipschitz numbers and fine differentiability on metric measure spaces
    Panu Lahti, Yuxin Li
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv

Cite

BibTeXRIS
BibTeX
@misc{thale2026realising,
  title = {{Realising atomless laws as distance distributions on metric measure spaces}},
  author = {Christoph Thäle and Philipp Tuchel},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.28330},
  eprint = {2608.28330},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.28330},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Realising atomless laws as distance distributions on metric measure spaces
AU  - Christoph Thäle
AU  - Philipp Tuchel
PY  - 2026
DA  - 2026/08/28
PB  - arXiv
DO  - 10.48550/arxiv.2608.28330
UR  - https://arxiv.org/abs/2608.28330
AB  - We study which probability measures on [0,∞) can occur as the distribution of the distance between two independent points sampled from a complete separable metric space equipped with a Borel probability measure. We prove that an atomless Borel probability measure can be realised in this way if and only if its support contains the origin. This settles the conjecture of Aldous, Blanc, and Curien for absolutely continuous laws and also covers atomless measures that are singular with respect to Lebesgue measure. Moreover, for every L>1, we show that the realising metric may be chosen L-bi-Lipschitz equivalent to an ultrametric. The proof constructs the space from compact components whose mutual distances produce prescribed parts of the measure, while distances within the components are assigned to smaller scales.
ER  -