Schramm–Loewner evolution

In probability theory, the Schramm–Loewner evolution with parameter κ, also known as stochastic Loewner evolution (SLEκ), is a family of random planar curves that have been proven to be the scaling limit of a variety of two-dimensional lattice models in statistical mechanics.

Given a parameter κ and a domain U in the complex plane, it gives a family of random curves in U, with κ controlling how much the curve turns.

There are two main variants of SLE, chordal SLE which gives a family of random curves from two fixed boundary points, and radial SLE, which gives a family of random curves from a fixed boundary point to a fixed interior point.

These curves are defined to satisfy conformal invariance and a domain Markov property.

It was discovered by Oded Schramm (2000) as a conjectured scaling limit of the planar uniform spanning tree (UST) and the planar loop-erased random walk (LERW) probabilistic processes, and developed by him together with Greg Lawler and Wendelin Werner in a series of joint papers.

Besides UST and LERW, the Schramm–Loewner evolution is conjectured or proven to describe the scaling limit of various stochastic processes in the plane, such as critical percolation, the critical Ising model, the double-dimer model, self-avoiding walks, and other critical statistical mechanics models that exhibit conformal invariance.

The main idea is that the conformal invariance and a certain Markov property inherent in such stochastic processes together make it possible to encode these planar curves into a one-dimensional Brownian motion running on the boundary of the domain (the driving function in Loewner's differential equation).

This way, many important questions about the planar models can be translated into exercises in Itô calculus.

Indeed, several mathematically non-rigorous predictions made by physicists using conformal field theory have been proven using this strategy.

is a simply connected, open complex domain not equal to

, then it satisfies a differential equation found by Loewner (1923, p. 121) in his work on the Bieberbach conjecture.

is the upper half plane the Loewner equation differs from this by changes of variable and is The driving function

be the upper half plane and consider an SLE0, so the driving function

is thus identically zero almost surely and Schramm–Loewner evolution is the random curve γ given by the Loewner equation as in the previous section, for the driving function where B(t) is Brownian motion on the boundary of D, scaled by some real κ.

There are two versions of SLE, using two families of curves, each depending on a non-negative real parameter κ: SLE depends on a choice of Brownian motion on the boundary of the domain, and there are several variations depending on what sort of Brownian motion is used: for example it might start at a fixed point, or start at a uniformly distributed point on the unit circle, or might have a built in drift, and so on.

The parameter κ controls the rate of diffusion of the Brownian motion, and the behavior of SLE depends critically on its value.

The two domains most commonly used in Schramm–Loewner evolution are the upper half plane and the unit disk.

Although the Loewner differential equation in these two cases look different, they are equivalent up to changes of variables as the unit disk and the upper half plane are conformally equivalent.

However a conformal equivalence between them does not preserve the Brownian motion on their boundaries used to drive Schramm–Loewner evolution.

(see Bauer & Bernard (2002a) Bauer & Bernard (2002b)) Beffara (2008) showed that the Hausdorff dimension of the paths (with probability 1) is equal to min(2, 1 + κ/8).

The probability of chordal SLEκ γ being on the left of fixed point

This was derived by using the martingale property of and Itô's lemma to obtain the following partial differential equation for

, which was used in the construction of the harmonic explorer,[2] and for κ = 6, we obtain Cardy's formula, which was used by Smirnov to prove conformal invariance in percolation.

[3] Lawler, Schramm & Werner (2001b) used SLE6 to prove the conjecture of Mandelbrot (1982) that the boundary of planar Brownian motion has fractal dimension 4/3.

Critical percolation on the triangular lattice was proved to be related to SLE6 by Stanislav Smirnov.

[4] Combined with earlier work of Harry Kesten,[5] this led to the determination of many of the critical exponents for percolation.

[6] This breakthrough, in turn, allowed further analysis of many aspects of this model.

[7][8] Loop-erased random walk was shown to converge to SLE2 by Lawler, Schramm and Werner.

The related random Peano curve outlining the uniform spanning tree was shown to converge to SLE8.

[9] Rohde and Schramm showed that κ is related to the fractal dimension of a curve by the following relation Computer programs (Matlab) are presented in this GitHub repository to simulate Schramm Loewner Evolution planar curves.