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Realising atomless laws as distance distributions on metric measure spaces — arXiv:2608.28330

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.28330
License (as archived): CC0-1.0
Archived: 2026-10-17T11:18:05.764Z
SHA-256 of the archived text: e0c4ee3ea802fec3901e1b09f70826344409a5302ea0b1f5497753c9c9604c17
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How to cite

Realising atomless laws as distance distributions on metric measure spaces — arXiv:2608.28330 — arXiv (en), https://arxiv.org/abs/2608.28330 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_5b2d2bfddeff (Metraipolis archive dc_5b2d2bfddeff, sha256 e0c4ee3ea802fec3901e1b09f70826344409a5302ea0b1f5497753c9c9604c17)

BibTeX

@misc{dc5b2d2bfddeff,
  title = {Realising atomless laws as distance distributions on metric measure spaces — arXiv:2608.28330},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.28330},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_5b2d2bfddeff; sha256 e0c4ee3ea802fec3901e1b09f70826344409a5302ea0b1f5497753c9c9604c17},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.28330. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We study which probability measures on $[0,\infty)$ 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.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-31 in math.PR).
Categorie arXiv: Probability (math.PR) ; Metric Geometry (math.MG).
Pagina del lavoro: https://arxiv.org/abs/2608.28330 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-17T11:05:49.685Z dalla pagina https://arxiv.org/catchup/math.PR/2026-08-31?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).