Communications in Mathematical Physics

, Volume 366, Issue 2, pp 469–536 | Cite as

Global and Local Multiple SLEs for \({\kappa \leq 4}\) and Connection Probabilities for Level Lines of GFF

  • Eveliina Peltola
  • Hao WuEmail author


This article pertains to the classification of multiple Schramm–Loewner evolutions (SLE). We construct the pure partition functions of multiple \({{\rm SLE}_\kappa}\) with \({\kappa \in (0,4]}\) and relate them to certain extremal multiple SLE measures, thus verifying a conjecture from Bauer et al. (J Stat Phys 120(5–6):1125–1163, 2005) and Kytölä and Peltola (Commun Math Phys 346(1):237–292, 2016). We prove that the two approaches to construct multiple SLEs—the global, configurational construction of Kozdron and Lawler (Universality and renormalization, vol 50 of Fields institute communications. American Mathematical Society, Providence, 2007) and Lawler (J Stat Phys 134(5–6): 813-837, 2009) and the local, growth process construction of Bauer et al. (2005), Dubédat (Commun Pure Appl Math 60(12):1792–1847, 2007), Graham (J Stat Mech Theory 2007(3):P03008, 2007) and Kytölä and Peltola (2016)—agree. The pure partition functions are closely related to crossing probabilities in critical statistical mechanics models. With explicit formulas in the special case of \({\kappa = 4}\), we show that these functions give the connection probabilities for the level lines of the Gaussian free field (GFF) with alternating boundary data. We also show that certain functions, known as conformal blocks, give rise to multiple SLE4 that can be naturally coupled with the GFF with appropriate boundary data.


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We thank V. Beffara, G. Lawler, and W. Qian for helpful discussions on multiple SLEs. We thank M. Russkikh for useful discussions on (double-)dimer models and B. Duplantier, S. Flores, A. Karrila, K. Kytölä, and A. Sepulveda for interesting, useful, and stimulating discussions. Part of this work was completed during H.W.’s visit at the IHES, which we cordially thank for hospitality. Finally, we are grateful to the referee for careful comments on the manuscript.


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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Section de MathématiquesUniversité de GenèveGenevaSwitzerland
  2. 2.Yau Mathematical Sciences CenterTsinghua UniversityBeijingChina

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