Basic properties of SLE

Abstract

SLEt<; is a random growth process based on Loewner's equation with driving parameter a one-dimensional Brownian motion running with speed K,. This process is intimately connected with scaling limits of percolation clusters and with the outer boundary of Brownian motion, and is conjectured to correspond to scaling limits of several other discrete processes in two dimensions.The present paper attempts a first systematic study of SLE.

Imaginary geometry IV: interior rays, whole-plane reversibility, and space-filling trees

“…The first proof that, when κ > 8, the time reversal of an SLE κ (ρ 1 ; ρ 2 ) process is a process that belongs to the same family. It has been known for some time [29] that SLE κ itself should not have time-reversal symmetry when κ > 8. However the fact that its time reversal can be described by an SLE κ (ρ 1 ; ρ 2 ) process was not known, or even conjectured, before the current work.…”

Connection Probabilities for Conformal Loop Ensembles

“…It has been conjectured (see for instance [16,20,43]), based on the identification between exponents, probabilities or dimensions that one can rigorously compute for SLE processes on the one hand and the corresponding quantities that had been previously predicted using methods from theoretical physics (conformal field theory, quantum gravity or Coulomb gas methods, see for instance [6,40]) for the asymptotic behavior of the discrete models on the other hand, that the O(N ) models have a non-trivial and conformally invariant scaling limit for all N ∈ (0, 2] and that this scaling limit should be related to SLE κ curves for N = −2 cos(4π/κ), where κ ∈ (8/3, 4] if one considers the dilute O(N ) model and where κ ∈ (4,8) if one considers the dense O(N ) model. Similarly, the scaling limit of the critical FK(q)-percolation model interfaces should be non-trivial for q ∈ (0, 4] and described by SLE κ curves, for √ q = −2 cos(4π/κ), where κ ∈ [4, 8) (mind of course that cos(4π/κ) is negative for all κ ∈ (8/3, 8)).…”

“…More precisely, the argument was the following: If these models have a conformally invariant scaling limit, then it must be described by an SLE κ for some κ. In order to identify which value of κ is the right one, the idea in those papers was to match some computation of probabilities of events for SLE (or of critical exponents, or of dimensions) with the corresponding values that were predicted to be the correct ones for the scaling limit of the discrete model, based on theoretical physics considerations (see for instance [43] for the FK(q) conjecture based on the physics dimension predictions; another approach is related to the discrete parafermionic observable for FK models-see [53] or [13] for a detailed discussion and more references-where the spin is defined as σ = 1−(2/π ) arccos( √ q/2), and that is conjectured to correspond in the scaling limit to some SLE martingale, see for instance [61]). In the present paper, we identify the candidate value of κ using a feature that is rigorously known to hold in the discrete model (and therefore in its scaling limit, if it exists).…”

“…10). The probability of D(ε) can be explicitly computed (it is in fact the formula that was already used by Schramm [44] in his argument mentioned at the beginning of the introduction); it is a generalization of Cardy's formula for SLE κ almost identical to that determined in [27]-see for instance [43,Lemma 6.6] or [57,Section 3]): Fig. 11.…”

Conformally invariant scaling limits: an overview and a collection of problems

“…It has been proven [76], [59] that with probability 1 there is a (unique) random continuous path γ (t) such that for each t ≥ 0 the domain of definition of g t is the unbounded component of H \ γ ([0, t] If D is a simply connected domain in the plane and a, b ∈ ∂D are two distinct points (or rather prime ends), then chordal SLE from a to b in D is defined as the image of γ under a conformal map from H to D taking 0 to a and ∞ to b. Though the map is not unique, the choice of the map does not effect the law of the SLE in D. This follows from the easily verified fact that up to a rescaling of time, the law of the SLE path is invariant under scaling by a positive real constant, as is the case for Brownian motion.…”