Pith Number

pith:KEPWWYGI

pith:2026:KEPWWYGIFQ4DFO6C7T3D7G5PHM

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Generalized intersection exponents and local cut points for three-dimensional Brownian loop soup

Ruixuan Li, Xinyi Li, Yifan Gao

Generalized intersection exponents exist for the three-dimensional Brownian loop soup and relate to the dimension of its local cut points.

arxiv:2605.17494 v1 · 2026-05-17 · math.PR

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Claims

C1strongest claim

We establish the existence of generalized intersection exponents (GIE) and prove an up-to-constants estimate for these probabilities by means of a separation lemma tailored to this setting. We also relate the Hausdorff dimension of the set of local cut points of the three-dimensional Brownian loop soup to the GIE, and show that the GIE is continuous at intensity zero, where it reduces to the classical Brownian intersection exponent. In particular, this implies that, for sufficiently small intensity parameters, the set of local cut points has Hausdorff dimension strictly larger than 1.

C2weakest assumption

The separation lemma tailored to the loop-soup setting holds and supplies the up-to-constants bounds needed to define the generalized intersection exponents; without it the existence claim and the dimension relation cannot be established from the non-intersection probabilities.

C3one line summary

The authors establish generalized intersection exponents for the 3D Brownian loop soup, prove an up-to-constants estimate via a separation lemma, relate them to the Hausdorff dimension of local cut points, and show continuity at zero intensity implying dimension strictly larger than 1 for small sub-

References

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[1] K. Burdzy and G. F. Lawler. Nonintersection exponents for Brownian paths. II. Estimates and applications to a random fractal. Ann. Probab., 18(3):981–1009, 1990 1990

[2] arXiv preprint arXiv:2406.02397 , year= 2024

[3] Z. Cai and J. Ding. Heterochromatic two-arm probabilities for metric graph Gaussian free ﬁelds. arXiv preprint arXiv:2510.20492 , 2025 2025

[4] Z. Cai and J. Ding. On the gap between cluster dimensions of loop soups on R3 and the metric graph of Z3. arXiv preprint arXiv:2510.20526 , 2025 2025

[5] Z. Cai and J. Ding. Separation and cut edge in macroscopic clusters for metric graph Gaussian free ﬁelds. arXiv preprint arXiv:2510.20516 , 2025 2025

Formal links

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Receipt and verification

First computed	2026-05-20T00:04:42.095240Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

511f6b60c82c3832bbc2fcf63f9baf3b31a170b2ead9d26685f663718abb1117

Aliases

arxiv: 2605.17494 · arxiv_version: 2605.17494v1 · doi: 10.48550/arxiv.2605.17494 · pith_short_12: KEPWWYGIFQ4D · pith_short_16: KEPWWYGIFQ4DFO6C · pith_short_8: KEPWWYGI

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curl -sH 'Accept: application/ld+json' https://pith.science/pith/KEPWWYGIFQ4DFO6C7T3D7G5PHM \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 511f6b60c82c3832bbc2fcf63f9baf3b31a170b2ead9d26685f663718abb1117

Canonical record JSON

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    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.PR",
    "submitted_at": "2026-05-17T15:05:53Z",
    "title_canon_sha256": "022ea897f5f64ffaa77e57fe3d904208660aa5b878e16c6b72f808569995f36a"
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