Global and local multiple SLEs for \(\kappa\leq4\) and connection probabilities for level lines of GFF. (English) Zbl 1422.60142
Summary: This article pertains to the classification of multiple Schramm-Loewner evolutions (SLE). We construct the pure partition functions of multiple \(\mathrm{SLE}_\kappa\) with \(\kappa\in(0,4]\) and relate them to certain extremal multiple SLE measures, thus verifying a conjecture from M. Bauer et al. [J. Stat. Phys. 120, No. 5–6, 1125–1163 (2005; Zbl 1094.82016)] and K. Kytölä and the first author [Commun. Math. Phys. 346, No. 1, 237–292 (2016; Zbl 1358.82012)]. We prove that the two approaches to construct multiple SLEs – the global, configurational construction of M. J. Kozdron and G. F. Lawler [Fields Inst. Commun. 50, 199–224 (2007; Zbl 1133.60023)] and G. F. Lawler [J. Stat. Phys. 134, No. 5–6, 813–837 (2009; Zbl 1168.82006)] and the local, growth process construction of Bauer et al. [loc. cit.], J. Dubédat [Commun. Pure Appl. Math. 60, No. 12, 1792–1847 (2007; Zbl 1137.82009)], K. Graham [“On multiple Schramm-Loewner evolutions”, J. Stat. Mech. Theory 2007, No. 3, 03008 (2007)] and Kytölä and Peltola [loc. cit.] – 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 \(\mathrm{SLE}_4\) that can be naturally coupled with the GFF with appropriate boundary data.
MSC:
| 60J67 | Stochastic (Schramm-)Loewner evolution (SLE) |
| 60G15 | Gaussian processes |
| 82B05 | Classical equilibrium statistical mechanics (general) |
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