Multiple SLE type scaling limits: from local to global

Abstract

We consider collections of $N$ chordal random curves obtained from a critical lattice model on a planar graph, in the limit when a fine-mesh graph approximates a simply-connected domain. We define and study candidates for such limits in terms of conformally invariant collections of random curves, generated via iterated Loewner equations. These curves are a natural ``domain Markov extension'' of the earlier introduced local multiple SLE initial segments to global multiple SLE curves. For realizing them as scaling limits, we provide two a priori results to guarantee the precompactness of the discrete random curves and to allow promoting a discrete domain Markov property to the scaling limit. These results essentially only take as input certain crossing conditions, very similar to those introduced by Kemppainen and Smirnov, and they allow the identification of scaling limits via the martingale strategy of classical SLE convergence proofs. The use of these results is exemplified with convergence proofs in various lattice models.

64 pages, 13 figures

References in corpus (14)

• The dimension of the SLE curves
• Multiple Schramm-Loewner Evolutions and Statistical Mechanics Martingales
• Boundary Partitions in Trees and Dimers
• Euler integrals for commuting SLEs
• On Multiple Schramm-Loewner Evolutions
• A solution space for a system of null-state partial differential equations 3
• Conformal invariance of boundary touching loops of FK Ising model
• SLE and Virasoro representations: Fusion
• Loop-erased random walk and Poisson kernel on planar graphs
• Convergence of radial loop-erased random walk in the natural parametrization
• Towards conformal invariance of 2D lattice models
• Limits of conformal images and conformal images of limits for planar random curves
• Basis for solutions of the Benoit & Saint-Aubin PDEs with particular asymptotic properties
• On the convergence of FK-Ising Percolation to SLE$(16/3,16/3-6)$

Cited by in corpus (7)

• UST branches, martingales, and multiple SLE(2)
• On the Uniqueness of Global Multiple SLEs
• Hypergeometric SLE with $\kappa =8$: Convergence of UST and LERW in Topological Rectangles
• Uniform Spanning Tree in Topological Polygons, Partition Functions for SLE(8), and Correlations in $c=-2$ Logarithmic CFT
• Multiple Ising interfaces in annulus and $2N$-sided radial SLE
• Loop-erased walks and random matrices
• Limits of radial multiple SLE and a Burgers-Loewner differential equation