---
tag: XW2BS4VG3P
kind: definition
name: Schramm–Loewner evolution
msc: [60J67, 30C35, 82B27]
defines: AQ75M2JTB2
uses: [51G4QJZ345, 22735FGJV6, X6XGGH5FPW, 0VHNKP1N3K]
sources:
  - url: https://en.wikipedia.org/wiki/Schramm%E2%80%93Loewner_evolution
---

## Definition

Let $\kappa > 0$, and let $(B_t)_{t \ge 0}$ be a standard one-dimensional [[Brownian motion|22735FGJV6]] with $B_0 = 0$.
Let $\mathbb{H}$ be the [[upper half-plane|X6XGGH5FPW]].
*Chordal $\mathrm{SLE}_\kappa$* in $\mathbb{H}$ from $0$ to $\infty$ is a random family of conformal maps $(g_t)_{t \ge 0}$ and hulls $(K_t)_{t \ge 0}$.
It is given by the chordal [[Loewner differential equation|51G4QJZ345]]
$$\frac{\partial}{\partial t} g_t(z) = \frac{2}{g_t(z) - \sqrt{\kappa}\, B_t}, \qquad g_0(z) = z,$$
with the driving function $U(t) = \sqrt{\kappa}\, B_t$.
Let $D \ne \C$ be a simply connected domain with two distinct boundary points $a$ and $b$.
Let $\phi$ be a [[conformal map|0VHNKP1N3K]] of $\mathbb{H}$ onto $D$ that sends $0$ to $a$ and $\infty$ to $b$.
Chordal $\mathrm{SLE}_\kappa$ in $D$ from $a$ to $b$ is the image under $\phi$ of chordal $\mathrm{SLE}_\kappa$ in $\mathbb{H}$.

## Remarks

The definition in $D$ does not depend on the choice of $\phi$, because two choices differ by a scaling $z \mapsto cz$ of $\mathbb{H}$ with $c > 0$.
Brownian scaling shows that such a scaling only changes the time parameter of the hulls.
Almost surely the hulls are generated by a continuous curve $\gamma$: $\mathbb{H} \setminus K_t$ is the unbounded component of $\mathbb{H} \setminus \gamma[0, t]$.
This was proved by Rohde and Schramm for $\kappa \ne 8$, and by Lawler, Schramm and Werner for $\kappa = 8$.
The curve is simple for $\kappa \le 4$, touches itself for $4 < \kappa < 8$, and fills space for $\kappa \ge 8$.
Example: for $\kappa = 2$, $\mathrm{SLE}_2$ is the scaling limit of loop-erased random walk, a theorem of Lawler, Schramm and Werner.
For $\kappa = 0$ the driving function is $0$ and the hulls are deterministic vertical segments; this degenerate case is sometimes included as $\mathrm{SLE}_0$.
Radial $\mathrm{SLE}_\kappa$ is defined in the same way from the radial Loewner equation.