Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Conditioning a Brownian loop-soup cluster on a portion of its boundary

Wei Qian

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We show that if one conditions a cluster in a Brownian loop-soup $L$ (of any intensity) in a two-dimensional domain by a portion $\partial $ of its outer boundary, then in the remaining domain, the union of all the loops of $L$ that touch $\partial $ satisfies the conformal restriction property while the other loops in $L$ form an independent loop-soup. This result holds when one discovers $\partial $ in a natural Markovian way, such as in the exploration procedures that have been defined in order to actually construct the Conformal Loop Ensembles as outer boundaries of loop-soup clusters. This result implies among other things that a phase transition occurs at $c=14/15$ for the connectedness of the loops that touch $\partial $.

Our results can be viewed as an extension of some of the results in our paper (J. Eur. Math. Soc. (2019) to appear) in the following two directions: There, a loop-soup cluster was conditioned on its entire outer boundary while we discover here only part of this boundary. And, while it was explained in (J. Eur. Math. Soc. (2019) to appear) that the strong decomposition using a Poisson point process of excursions that we derived there should be specific to the case of the critical loop-soup, we show here that in the subcritical cases, a weaker property involving the conformal restriction property nevertheless holds.


Dans le présent article, nous étudions certaines propriétés des amas de lacets browniens, dans une soupe de lacets browniens d’intensité $c$ dans un domaine du plan, pour toute intensité $c\le 1$.

Notre principal résultat dit que si l’on découvre de manière Markovienne une portion $\partial $ du bord extérieur d’un tel amas, alors dans le domaine restant, la loi conditionnelle de l’union de tous les lacets dans $L$ qui touchent $\partial $ satisfait la propriété de restriction conforme tandis que les autres lacets dans $L$ forment une soupe de lacets indépendante. Ceci implique en particulier l’existence d’une transition de phase à $c=14/15$ pour la connectivité de l’ensemble des lacets qui touchent $\partial $.

Nos résultats constituent une extension de certains résultats de notre papier (J. Eur. Math. Soc. (2019) to appear) dans les deux directions suivantes: Dans (J. Eur. Math. Soc. (2019) to appear), un cluster de lacets est conditionné par son bord extérieur entier tandis que nous découvrons ici seulement une partie de ce bord. En outre, dans (J. Eur. Math. Soc. (2019) to appear), nous expliquons que la description que nous donnons de l’ensemble des lacets qui touchent ce bord via un processus ponctuel de Poisson d’excursions est spécifique au cas de la soupe de lacets critique ($c=1$), nous montrons ici que dans les cas sous-critiques $c<1$, une propriété plus faible de restriction conforme reste néanmoins vraie.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 55, Number 1 (2019), 314-340.

Received: 6 March 2017
Revised: 25 September 2017
Accepted: 11 January 2018
First available in Project Euclid: 18 January 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J65: Brownian motion [See also 58J65] 60J67: Stochastic (Schramm-)Loewner evolution (SLE) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Brownian loop-soups Conformal loop ensembles Schramm–Loewner evolution Conformal restriction


Qian, Wei. Conditioning a Brownian loop-soup cluster on a portion of its boundary. Ann. Inst. H. Poincaré Probab. Statist. 55 (2019), no. 1, 314--340. doi:10.1214/18-AIHP883.

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  • [1] J. Aru, A. Sepúlveda and W. Werner. On bounded-type thin local sets of the two-dimensional Gaussian free field. J. Inst. Math. Jussieu. (2017) 1–28.
  • [2] S. Benoist and C. Hongler. The scaling limit of critical Ising interfaces is $\operatorname{CLE}(3)$. Ann. Probab. To appear, 2019. Available at arXiv:1604.06975.
  • [3] K. Burdzy. Cut points on Brownian paths. Ann. Probab. 17 (3) (1989) 1012–1036.
  • [4] K. Burdzy and G. F. Lawler. Nonintersection exponents for Brownian paths. II. Estimates and applications to a random fractal. Ann. Probab. 18 (3) (1990) 981–1009.
  • [5] D. Friedan, Z. Qiu and S. Shenker. Conformal invariance, unitarity, and critical exponents in two dimensions. Phys. Rev. Lett. 52 (1984) 1575–1578.
  • [6] G. F. Lawler, O. Schramm and W. Werner. Conformal restriction: The chordal case. J. Amer. Math. Soc. 16 (4) (2003) 917–955 (electronic).
  • [7] G. F. Lawler and J. A. Trujillo Ferreras. Random walk loop soup. Trans. Amer. Math. Soc. 359 (2) (2007) 767–787 (electronic).
  • [8] G. F. Lawler and W. Werner. The Brownian loop soup. Probab. Theory Related Fields 128 (4) (2004) 565–588.
  • [9] T. Lupu. Convergence of the two-dimensional random walk loop soup clusters to CLE. J. Eur. Math. Soc. To appear, 2019. Available at arXiv:1502.06827.
  • [10] J. Miller and S. Sheffield. CLE(4) and the Gaussian free field. In preparation.
  • [11] J. Miller, S. Sheffield and W. Werner. CLE percolations. Forum Math., Pi 5 (2017) Article ID e4. DOI:10.1017/fmp.2017.5.
  • [12] J. Miller, S. Sheffield and W. Werner. Non-simple SLE curves are not determined by their range. Preprint. Available at arXiv:1609.04799.
  • [13] W. Qian. Conformal restriction: The trichordal case. Probab. Theory Related Fields 171 (3–4) (2018) 709–774.
  • [14] W. Qian and W. Werner. Decomposition of Brownian loop-soup clusters. J. Eur. Math. Soc. To appear, 2019. Available at arXiv:1509.01180.
  • [15] S. Sheffield. Exploration trees and conformal loop ensembles. Duke Math. J. 147 (1) (2009) 79–129.
  • [16] S. Sheffield and W. Werner. Conformal loop ensembles: The Markovian characterization and the loop-soup construction. Ann. of Math. (2) 176 (3) (2012) 1827–1917.
  • [17] T. van de Brug, F. Camia and M. Lis. Random walk loop soups and conformal loop ensembles. Probab. Theory Related Fields 166 (1–2) (2016) 553–584.
  • [18] W. Wendelin. SLEs as boundaries of clusters of Brownian loops. C. R. Math. Acad. Sci. Paris 337 (7) (2003) 481–486.
  • [19] W. Wendelin. Conformal restriction and related questions. Probab. Surv. 2 (2005) 145–190.
  • [20] W. Wendelin. The conformally invariant measure on self-avoiding loops. J. Amer. Math. Soc. 21 (1) (2008) 137–169.
  • [21] W. Werner and H. Wu. From $\mathrm{CLE}(\kappa)$ to $\mathrm{SLE}(\kappa ,\rho)$’s. Electron. J. Probab. 18 (36) (2013) 20.
  • [22] W. Werner and H. Wu. On conformally invariant CLE explorations. Comm. Math. Phys. 320 (3) (2013) 637–661.
  • [23] H. Wu. Conformal restriction: The radial case. Stochastic Process. Appl. 125 (2) (2015) 552–570.