MULTIPLE POINTS ON THE BOUNDARIES OF BROWNIAN LOOP-SOUP CLUSTERS

Yifan GAO; Xinyi LI; Wei QIAN

doi:10.1214/25-AOP1765

MULTIPLE POINTS ON THE BOUNDARIES OF BROWNIAN LOOP-SOUP CLUSTERS

Yifan GAO^*
, Xinyi LI
, Wei QIAN

^*Corresponding author for this work

Department of Mathematics

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

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Abstract

For a Brownian loop soup with intensity c ∈ (0, 1] in the unit disk, we show that almost surely, the set of simple (resp., double) points on any portion of boundary of any of its clusters has Hausdorff dimension 2 − ξ_c(2) (resp., 2 − ξ_c(4)), where ξ_c(k) is the generalized disconnection exponent computed in (Probab. Theory Related Fields 179 (2021) 117–164). As a consequence, when the dimension is positive, such points are a.s. dense on every boundary of every cluster. There are a.s. no triple points on the cluster boundaries.

As an intermediate result, we establish a separation lemma for Brownian loop soups, which is a powerful tool for obtaining sharp estimates on non-intersection and nondisconnection probabilities in the setting of loop soups. In particular, it allows us to define a family of generalized intersection exponents ξ_c(k, λ), and show that ξ_c(k) is the limit as λ ↘ 0 of ξ_c(k, λ). © 2026 Institute of Mathematical Statistics.

Original language	English
Pages (from-to)	216–268
Number of pages	53
Journal	Annals of Probability
Volume	54
Issue number	1
DOIs	https://doi.org/10.1214/25-AOP1765
Publication status	Published - Jan 2026

Funding

YG and XL acknowledge the support by National Key R&D Program of China (No. 2021YFA1002700 and No. 2020YFA0712900) and NSFC (No. 12071012). YG and WQ acknowledge the support by National Key R&D Program of China (No. 2023YFA1010700) and a grant from City University of Hong Kong (Project No. 7200745). WQ acknowledges the support by the National Science Foundation under Grant No. 1440140, while she was in residence at the Simons Laufer Mathematical Sciences Institute in Berkeley, California, during the spring semester of 2022.

Research Keywords

Brownian loop soup
separation lemma
generalized disconnection and intersection exponents

Publisher's Copyright Statement

COPYRIGHT TERMS OF DEPOSITED FINAL PUBLISHED VERSION FILE: © Institute of Mathematical Statistics, 2026 GAO, Y., LI, X., & QIAN, W. (2026). MULTIPLE POINTS ON THE BOUNDARIES OF BROWNIAN LOOP-SOUP CLUSTERS. Annals of Probability, 54(1), 216–268. https://doi.org/10.1214/25-AOP1765

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abstract = "For a Brownian loop soup with intensity c ∈ (0, 1] in the unit disk, we show that almost surely, the set of simple (resp., double) points on any portion of boundary of any of its clusters has Hausdorff dimension 2 − ξc(2) (resp., 2 − ξc(4)), where ξc(k) is the generalized disconnection exponent computed in (Probab. Theory Related Fields 179 (2021) 117–164). As a consequence, when the dimension is positive, such points are a.s. dense on every boundary of every cluster. There are a.s. no triple points on the cluster boundaries. As an intermediate result, we establish a separation lemma for Brownian loop soups, which is a powerful tool for obtaining sharp estimates on non-intersection and nondisconnection probabilities in the setting of loop soups. In particular, it allows us to define a family of generalized intersection exponents ξc(k, λ), and show that ξc(k) is the limit as λ ↘ 0 of ξc(k, λ). {\textcopyright} 2026 Institute of Mathematical Statistics.",

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author = "Yifan GAO and Xinyi LI and Wei QIAN",

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AU - GAO, Yifan

AU - LI, Xinyi

AU - QIAN, Wei

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Y1 - 2026/1

N2 - For a Brownian loop soup with intensity c ∈ (0, 1] in the unit disk, we show that almost surely, the set of simple (resp., double) points on any portion of boundary of any of its clusters has Hausdorff dimension 2 − ξc(2) (resp., 2 − ξc(4)), where ξc(k) is the generalized disconnection exponent computed in (Probab. Theory Related Fields 179 (2021) 117–164). As a consequence, when the dimension is positive, such points are a.s. dense on every boundary of every cluster. There are a.s. no triple points on the cluster boundaries. As an intermediate result, we establish a separation lemma for Brownian loop soups, which is a powerful tool for obtaining sharp estimates on non-intersection and nondisconnection probabilities in the setting of loop soups. In particular, it allows us to define a family of generalized intersection exponents ξc(k, λ), and show that ξc(k) is the limit as λ ↘ 0 of ξc(k, λ). © 2026 Institute of Mathematical Statistics.

AB - For a Brownian loop soup with intensity c ∈ (0, 1] in the unit disk, we show that almost surely, the set of simple (resp., double) points on any portion of boundary of any of its clusters has Hausdorff dimension 2 − ξc(2) (resp., 2 − ξc(4)), where ξc(k) is the generalized disconnection exponent computed in (Probab. Theory Related Fields 179 (2021) 117–164). As a consequence, when the dimension is positive, such points are a.s. dense on every boundary of every cluster. There are a.s. no triple points on the cluster boundaries. As an intermediate result, we establish a separation lemma for Brownian loop soups, which is a powerful tool for obtaining sharp estimates on non-intersection and nondisconnection probabilities in the setting of loop soups. In particular, it allows us to define a family of generalized intersection exponents ξc(k, λ), and show that ξc(k) is the limit as λ ↘ 0 of ξc(k, λ). © 2026 Institute of Mathematical Statistics.

KW - Brownian loop soup

KW - separation lemma

KW - generalized disconnection and intersection exponents

UR - https://www.webofscience.com/wos/woscc/full-record/WOS:001691652400006

U2 - 10.1214/25-AOP1765

DO - 10.1214/25-AOP1765

M3 - RGC 21 - Publication in refereed journal

SN - 0091-1798

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SP - 216

EP - 268

JO - Annals of Probability

JF - Annals of Probability

IS - 1

ER -

MULTIPLE POINTS ON THE BOUNDARIES OF BROWNIAN LOOP-SOUP CLUSTERS

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Research Keywords

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