Global and Local Multiple SLEs for $\kappa \le 4$ and Connection Probabilities for Level Lines of GFF

Abstract

This article pertains to the classification of multiple Schramm-Loewner evolutions (SLE). We construct the pure partition functions of multiple SLE$(\kappa )$ with $\kappa \in (0,4]$ and relate them to certain extremal multiple SLE measures, thus verifying a conjecture from [BBK05, KP16]. We prove that the two approaches to construct multiple SLEs - the global, configurational construction of [KL07, Law09a] and the local, growth process construction of [BBK05, Dub07, Gra07, KP16] - 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 SLE(4) that can be naturally coupled with the GFF with appropriate boundary data.

arXiv comment59 pages, 12 figures. Final version. To appear at Comm. Math. Phys

References in corpus (23)

• Conformal restriction: the chordal case
• Conformal Field Theories of Stochastic Loewner Evolutions
• Multiple Schramm-Loewner Evolutions and Statistical Mechanics Martingales
• Commutation relations for SLE
• Boundary Partitions in Trees and Dimers
• Euler integrals for commuting SLEs
• Reversibility of chordal SLE
• Conformal restriction, highest-weight representations and SLE
• On Conformal Field Theory and Stochastic Loewner Evolution
• On Multiple Schramm-Loewner Evolutions
• The scaling limits of planar LERW in finitely connected domains
• Conformal transformations and the SLE partition function martingale
• A solution space for a system of null-state partial differential equations 3
• Smirnov's observable for free boundary conditions, interfaces and crossing probabilities
• Pure partition functions of multiple SLEs
• A solution space for a system of null-state partial differential equations 2
• SLE and Virasoro representations: Fusion
• Virasoro Module Structure of Local Martingales of SLE Variants
• A solution space for a system of null-state partial differential equations 1
• A formula for crossing probabilities of critical systems inside polygons
• A solution space for a system of null-state partial differential equations 4
• Reversed radial SLE and the Brownian loop measure
• On the smoothness of the partition function for multiple Schramm-Loewner evolutions

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