Basic properties of SLE

  title={Basic properties of SLE},
  author={Steffen Rohde and Oded Schramm},
  journal={Annals of Mathematics},
SLEκ is a random growth process based on Loewner’s equation with driving parameter a one-dimensional Brownian motion running with speed κ. 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. 

Figures from this paper

Cadlag curves of SLE driven by Levy processes
Schramm Loewner Evolutions (SLE) are random increasing hulls defined through the Loewner equation driven by Brownian motion. It is known that the increasing hulls are generated by continuous curves.Expand
Defining SLE in multiply connected domains with the Brownian loop measure
We define the Schramm-Loewner evolution (SLE) in multiply connected domains for kappa \leq 4 using the Brownian loop measure. We show that in the case of the annulus, this is the same measureExpand
Poisson point processes, excursions and stable processes in two-dimensional structures
Ito's contributions lie at the root of stochastic calculus and of the theory of excursions. These ideas are also very useful in the study of conformally invariant two-dimensional structures, viaExpand
The Intersection Probability of Brownian Motion and SLEκ
By using excursion measure Poisson kernel method, we obtain a second-order differential equation of the intersection probability of Brownian motion and . Moreover, we find a transformation such thatExpand
A level line of the Gaussian free field with measure-valued boundary conditions
In this article, we construct samples of SLE-like curves out of samples of CLE and Poisson point process of Brownian excursions. We show that the law of these curves depends continuously on theExpand
On The Brownian Loop Measure
In 2003 Lawler and Werner introduced the Brownian loop measure and studied some of its properties. In 2006 Cardy and Gamsa predicted a formula for the total mass that the Brownian loop measureExpand
The dimension of the SLE curves
Let γ be the curve generating a Schramm–Loewner Evolution (SLE) process, with parameter κ ≥ 0. We prove that, with probability one, the Haus-dorff dimension of γ is equal to Min(2, 1 + κ/8).Expand
Conformal restriction of Brownian excursion with darning
  • J. Ma, S. Lan
  • Mathematics
  • Complex Variables and Elliptic Equations
  • 2018
ABSTRACT Let be a two-dimensional Brownian excursion with darning on a finitely connected domain. Using Koebe's theorem and conformal invariance of Brownian motion with darning we derive that aExpand
Large deviations of radial SLE∞
We derive the large deviation principle for radial Schramm-Loewner evolution (SLE) on the unit disk with parameter κ → ∞. Restricting to the time interval [0, 1], the good rate function is finiteExpand
On the Rate of Convergence of Loop-Erased Random Walk to SLE2
We derive a rate of convergence of the Loewner driving function for a planar loop-erased random walk to Brownian motion with speed 2 on the unit circle, the Loewner driving function for radial SLE2.Expand


Critical Exponents, Conformal Invariance and Planar Brownian Motion
In this review paper, we first discuss some open problems related to two-dimensional self-avoiding paths and critical percolation. We then review some closely related results (joint work with GregExpand
SLE(κ,ρ) martingales and duality
Various features of the two-parameter family of Schramm-Loewner evolutions SLE(K, p) are studied. In particular, we derive certain restriction properties that lead to a strong duality conjecture,Expand
Critical percolation in the plane : I. Conformal invariance and Cardy's formula. II. Continuum scaling limit
We study scaling limits and conformal invariance of critical site percolation on triangular lattice. We show that some percolation-related quantities are harmonic conformal invariants, and calculateExpand
Values of Brownian intersection exponents, I: Half-plane exponents
Theoretical physics predicts that conformal invariance plays a crucial role in the macroscopic behavior of a wide class of two-dimensional models in statistical physics (see, e.g., [4], [6]). ForExpand
Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits
Abstract In this Note we study critical site percolation on triangular lattice. We introduce harmonic conformal invariants as scaling limits of certain probabilities and calculate their values. As aExpand
Conformal Invariance and Percolation
These lectures give an introduction to the methods of conformal field theory as applied to deriving certain results in two-dimensional critical percolation: namely the probability that there existsExpand
Values of Brownian intersection exponents III: Two-sided exponents
Abstract This paper determines values of intersection exponents between packs of planar Brownian motions in the half-plane and in the plane that were not derived in our first two papers. ForExpand
Aggregation in the Plane and Loewner's Equation
Abstract: We study an aggregation process which can be viewed as a deterministic analogue of the DLA model in the plane, or as a regularized version of the Hele-Shaw problem. The process is definedExpand
Scaling limits of loop-erased random walks and uniform spanning trees
AbstractThe uniform spanning tree (UST) and the loop-erased random walk (LERW) are strongly related probabilistic processes. We consider the limits of these models on a fine grid in the plane, as theExpand
Boundary Behaviour of Conformal Maps
1. Some Basic Facts.- 2. Continuity and Prime Ends.- 3. Smoothness and Corners.- 4. Distortion.- 5. Quasidisks.- 6. Linear Measure.- 7. Smirnov and Lavrentiev Domains.- 8. Integral Means.- 9. CurveExpand