Schramm–Loewner evolution

If ${\displaystyle D}$ is a simply connected, open complex domain not equal to ${\displaystyle \mathbb {C} }$, and ${\displaystyle \gamma }$ is a simple curve in ${\displaystyle D}$ starting on the boundary (a continuous function with ${\displaystyle \gamma (0)}$ on the boundary of ${\displaystyle D}$ and ${\displaystyle \gamma ((0,\infty ))}$ a subset of ${\displaystyle D}$), then for each ${\displaystyle t\geq 0}$, the complement ${\displaystyle D_{t}=D\setminus \gamma ([0,t])}$ of ${\displaystyle \gamma ([0,t])}$ is simply connected and therefore conformally isomorphic to ${\displaystyle D}$ by the Riemann mapping theorem. If ${\displaystyle f_{t}}$ is a suitable normalized isomorphism from ${\displaystyle D}$ to ${\displaystyle D_{t}}$, then it satisfies a differential equation found by Loewner (1923, p. 121) in his work on the Bieberbach conjecture. Sometimes it is more convenient to use the inverse function ${\displaystyle g_{t}}$ of ${\displaystyle f_{t}}$, which is a conformal mapping from ${\displaystyle D_{t}}$ to ${\displaystyle D}$.

In Loewner's equation, ${\displaystyle z\in D}$, ${\displaystyle t\geq 0}$, and the boundary values at time ${\displaystyle t=0}$ are ${\displaystyle f_{0}(z)=z}$ or ${\displaystyle g_{0}(z)=z}$. The equation depends on a driving function ${\displaystyle \zeta (t)}$ taking values in the boundary of ${\displaystyle D}$. If ${\displaystyle D}$ is the unit disk and the curve ${\displaystyle \gamma }$ is parameterized by "capacity", then Loewner's equation is

${\displaystyle {\frac {\partial f_{t}(z)}{\partial t}}=-zf_{t}^{\prime }(z){\frac {\zeta (t)+z}{\zeta (t)-z}}}$   or   ${\displaystyle {\dfrac {\partial g_{t}(z)}{\partial t}}=g_{t}(z){\dfrac {\zeta (t)+g_{t}(z)}{\zeta (t)-g_{t}(z)}}.}$

When ${\displaystyle D}$ is the upper half plane the Loewner equation differs from this by changes of variable and is

${\displaystyle {\frac {\partial f_{t}(z)}{\partial t}}={\frac {2f_{t}^{\prime }(z)}{\zeta (t)-z}}}$   or   ${\displaystyle {\dfrac {\partial g_{t}(z)}{\partial t}}={\dfrac {2}{g_{t}(z)-\zeta (t)}}.}$

The driving function ${\displaystyle \zeta }$ and the curve ${\displaystyle \gamma }$ are related by

${\displaystyle f_{t}(\zeta (t))=\gamma (t){\text{ or }}\zeta (t)=g_{t}(\gamma (t))}$

where ${\displaystyle f_{t}}$ and ${\displaystyle g_{t}}$ are extended by continuity.

Schramm–Loewner evolution is the random curve γ given by the Loewner equation as in the previous section, for the driving function

${\displaystyle \zeta (t)={\sqrt {\kappa }}B(t)}$

where B(t) is Brownian motion on the boundary of D, scaled by some real κ. In other words, Schramm–Loewner evolution is a probability measure on planar curves, given as the image of Wiener measure under this map.

In general the curve γ need not be simple, and the domain D_t is not the complement of γ([0,t]) in D, but is instead the unbounded component of the complement.

There are two versions of SLE, using two families of curves, each depending on a non-negative real parameter κ:

• Chordal SLE_κ, which is related to curves connecting two points on the boundary of a domain (usually the upper half plane, with the points being 0 and infinity).
• Radial SLE_κ, which is related to curves joining a point on the boundary of a domain to a point in the interior (often curves joining 1 and 0 in the unit disk).

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.

• For 0 ≤ κ < 4 the curve γ(t) is simple (with probability 1).
• For 4 < κ < 8 the curve γ(t) intersects itself and every point is contained in a loop but the curve is not space-filling (with probability 1).
• For κ ≥ 8 the curve γ(t) is space-filling (with probability 1).
• κ = 2 corresponds to the loop-erased random walk, or equivalently, branches of the uniform spanning tree.
• For κ = 8/3, SLE_κ has the restriction property and is conjectured to be the scaling limit of self-avoiding random walks. A version of it is the outer boundary of Brownian motion.
• κ = 3 is the limit of interfaces for the Ising model.
• κ = 4 corresponds to the path of the harmonic explorer and contour lines of the Gaussian free field.
• For κ = 6, SLE_κ has the locality property. This arises in the scaling limit of critical percolation on the triangular lattice and conjecturally on other lattices.
• κ = 8 corresponds to the path separating the uniform spanning tree from its dual tree.

When SLE corresponds to some conformal field theory, the parameter κ is related to the central charge c of the conformal field theory by

${\displaystyle c={\frac {(8-3\kappa )(\kappa -6)}{2\kappa }}.}$

Each value of c < 1 corresponds to two values of κ, one value κ between 0 and 4, and a "dual" value 16/κ greater than 4. (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 SLE_κ γ being on the left of fixed point ${\displaystyle x_{0}+iy_{0}=z_{0}\in \mathbb {H} }$ was computed by Schramm (2001a)[1]

${\displaystyle \mathbb {P} [\gamma {\text{ passes to the left }}z_{0}]={\frac {1}{2}}+{\frac {\Gamma ({\frac {4}{\kappa }})}{{\sqrt {\pi }}\,\Gamma ({\frac {8-\kappa }{2\kappa }})}}{\frac {x_{0}}{y_{0}}}\,_{2}F_{1}\left({\frac {1}{2}},{\frac {4}{\kappa }},{\frac {3}{2}},-\left({\frac {x_{0}}{y_{0}}}\right)^{2}\right)}$

where ${\displaystyle \Gamma }$ is the Gamma function and ${\displaystyle _{2}F_{1}(a,b,c,d)}$ is the hypergeometric function. This was derived by using the martingale property of

${\displaystyle h(x,y):=\mathbb {P} [\gamma {\text{ passes to the left }}x+iy]}$

and Itô's lemma to obtain the following partial differential equation for ${\displaystyle w:={\tfrac {x}{y}}}$

${\displaystyle {\frac {\kappa }{2}}\partial _{ww}h(w)+{\frac {4w}{w^{2}+1}}\partial _{w}h=0.}$

For κ = 4, the RHS is ${\displaystyle 1-{\tfrac {1}{\pi }}\arg(z_{0})}$, 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.[9] This allowed derivation of many quantitative properties of loop-erased random walk (some of which were derived earlier by Richard Kenyon[10]). 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

${\displaystyle d=1+{\frac {\kappa }{8}}.}$

Computer programs (Matlab) are presented in this GitHub repository to simulate Schramm Loewner Evolution planar curves.

• Lawler; Schramm; Werner (2001), Tutorial: SLE, Lawrence Hall of Science, University of California, Berkeley{{citation}}: CS1 maint: location missing publisher (link) ( video of MSRI lecture)
• Schramm, Oded (2001), Conformally Invariant Scaling Limits and SLE, MSRI (Slides from a talk.)