Contour lines of the two-dimensional discrete Gaussian free field

2009/01/01 by Oded Schramm, Scott Sheffield, Scott Sheffield⋆ · 19 citations
Mathematics · #Mathematical Dynamics and Fractals #Geometric Analysis and Curvature Flows #Stochastic processes and statistical mechanics

paper · pdf · doi:10.1007/s11511-009-0034-y

Abstract

We prove that the chordal contour lines of the discrete Gaussian free field converge to forms of SLE(4). Specifically, there is a constant λ > 0 such that when h is an interpolation of the discrete Gaussian free field on a Jordan domain—with boundary values −λ on one boundary arc and λ on the complementary arc—the zero level line of h joining the endpoints of these arcs converges to SLE(4) as the domain grows larger. If instead the boundary values are −a < 0 on the first arc and b > 0 on the complementary arc, then the convergence is to SLE(4; a/λ - 1, b/λ - 1), a variant of SLE(4).

Cited by

• The bulk one‐arm exponent for the CLE κ′ percolations
• Smirnov’s Observable for Free Boundary Conditions, Interfaces and Crossing Probabilities
• Fock space of local fields of the discrete GFF and its scaling limit bosonic CFT
• Rotational invariance in critical planar lattice models
• On the chemical distance exponent for the two-sided level-set of the 2D Gaussian free field
• Conformal field theory for annulus SLE: partition functions and martingale-observables
• Determinantal Point Processes, Stochastic Log-Gases, and Beyond
• Schramm-Loewner evolution contains a topological Sierpiński carpet when κ is close to 8
• Conformal Invariance of the FK-Ising Model on Lorentz-Maximal S-Embeddings
• Regularity of the SLE₄ uniformizing map and the SLE₈ trace
• Mating of trees for random planar maps and Liouville quantum gravity: a survey
• Connection probabilities in the double-dimer model -- the case of two\n connectivity patterns
• Conformally invariant random fields, Liouville quantum gravity measures, and random Paneitz operators on Riemannian manifolds of even dimension
• Coupling of radial multiple SLE with Gaussian free field, and the hydrodynamic limit
• Existence and uniqueness of the conformally covariant geodesic metric on non-simple conformal loop ensemble gaskets
• Full extremal process in four-dimensional membrane model
• The rate of convergence of harmonic explorer to SLE4
• Hypergeometric SLE with κ=8: Convergence of UST and LERW in Topological Rectangles
• Conformal Invariance of Boundary Touching Loops of FK Ising Model. [europepmc]

Related

• Contour lines of the two-dimensional discrete Gaussian free field
• A contour line of the continuum Gaussian free field
• A Gaussian free field approach to the natural parametrisation of SLE₄
• Hadamard's formula and couplings of SLEs with free field
• KPZ relation does not hold for the level lines and the SLE_κ flow lines of the Gaussian free field