In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, the Schramm–Loewner evolution with parameter ''κ'', also known as stochastic Loewner evolution (SLE
''κ''), 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
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
. Given a parameter ''κ'' and a domain in the complex plane ''U'', 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
In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group. The extension includes special conformal transformations and dilations. In three spatial plus one time dimensions, conformal symmetry ...
and a domain
Markov property
In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process. It is named after the Russian mathematician Andrey Markov. The term strong Markov property is similar to the Markov propert ...
.
It was discovered by as a conjectured scaling limit of the planar
uniform spanning tree
A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...
(UST) and the planar
loop-erased random walk
In mathematics, loop-erased random walk is a model for a random simple path with important applications in combinatorics, physics and quantum field theory. It is intimately connected to the uniform spanning tree, a model for a random tree. See als ...
(LERW) probabilistic processes, and developed by him together with
Greg Lawler
Gregory Francis Lawler (born July 14, 1955) is an American mathematician working in probability theory and best known for his work since 2000 on the Schramm–Loewner evolution.
He received his PhD from Princeton University in 1979 under the supe ...
and
Wendelin Werner
Wendelin Werner (born 23 September 1968) is a German-born French mathematician working on random processes such as self-avoiding random walks, Brownian motion, Schramm–Loewner evolution, and related theories in probability theory and mathematica ...
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 walk
In mathematics, a self-avoiding walk (SAW) is a sequence of moves on a lattice (a lattice path) that does not visit the same point more than once. This is a special case of the graph theoretical notion of a path. A self-avoiding polygon (SAP) ...
s, and other critical
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
models that exhibit conformal invariance. The SLE curves are the scaling limits of interfaces and other non-self-intersecting random curves in these models. The main idea is that the conformal invariance and a certain
Markov property
In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process. It is named after the Russian mathematician Andrey Markov. The term strong Markov property is similar to the Markov propert ...
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
Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance and stochastic differential equations.
The central ...
. Indeed, several mathematically non-rigorous predictions made by physicists using
conformal field theory
A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes ...
have been proven using this strategy.
The Loewner equation
If ''D'' is a
simply connected,
open
Open or OPEN may refer to:
Music
* Open (band), Australian pop/rock band
* The Open (band), English indie rock band
* ''Open'' (Blues Image album), 1969
* ''Open'' (Gotthard album), 1999
* ''Open'' (Cowboy Junkies album), 2001
* ''Open'' (YF ...
complex domain not equal to C, and ''γ'' is a simple curve in ''D'' starting on the boundary (a continuous function with ''γ''(0) on the boundary of ''D'' and ''γ''((0, ∞)) a subset of ''D''), then for each ''t'' ≥ 0, the complement ''D''
''t'' of ''γ''(
, ''t'' is simply connected and therefore
conformally isomorphic to ''D'' by the
Riemann mapping theorem
In complex analysis, the Riemann mapping theorem states that if ''U'' is a non-empty simply connected open subset of the complex number plane C which is not all of C, then there exists a biholomorphic mapping ''f'' (i.e. a bijective holomorphi ...
. If ''ƒ''
''t'' is a suitable normalized isomorphism from ''D'' to ''D''
''t'', then it satisfies a differential equation found by in his work on the
Bieberbach conjecture In complex analysis, de Branges's theorem, or the Bieberbach conjecture, is a theorem that gives a necessary condition on a holomorphic function in order for it to map the open unit disk of the complex plane injectively to the complex plane. It ...
.
Sometimes it is more convenient to use the inverse function ''g''
''t'' of ''ƒ''
''t'', which is a conformal mapping from ''D''
''t'' to ''D''.
In Loewner's equation, ''z'' is in the domain ''D'', ''t'' ≥ 0, and the boundary values at time ''t'' = 0 are ''ƒ''
0(''z'') = ''z'' or ''g''
0(''z'') = ''z''. The equation depends on a driving function ''ζ''(''t'') taking values in the boundary of ''D''. If ''D'' is the unit disk and the curve ''γ'' is parameterized by "capacity", then Loewner's equation is
:
or
When ''D'' is the upper half plane the Loewner equation differs from this by changes of variable and is
:
or
The driving function ''ζ'' and the curve ''γ'' are related by
:
where
and
are extended by continuity.
Example
Let ''D'' be the upper half plane and consider an SLE
0, so the driving function ''ζ'' is a Brownian motion of diffusivity zero. The function ''ζ'' is thus identically zero almost surely and
:
:
:
:
is the upper half-plane with the line from 0 to
removed.
Schramm–Loewner evolution
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''. 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''(
,''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 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 circle. Although the Loewner differential equation in these two cases look different, they are equivalent up to changes of variables as the unit circle 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.
Special values of ''κ''
*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
In mathematics, loop-erased random walk is a model for a random simple path with important applications in combinatorics, physics and quantum field theory. It is intimately connected to the uniform spanning tree, a model for a random tree. See als ...
, 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
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
.
*''κ'' = 3 is the limit of interfaces for the
Ising model
The Ising model () (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent ...
.
*''κ'' = 4 corresponds to the path of the harmonic explorer and contour lines of the
Gaussian free field
In probability theory and statistical mechanics, the Gaussian free field (GFF) is a Gaussian random field, a central model of random surfaces (random height functions). gives a mathematical survey of the Gaussian free field.
The discrete version ...
.
*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
In theoretical physics, a central charge is an operator ''Z'' that commutes with all the other symmetry operators. The adjective "central" refers to the center of the symmetry group—the subgroup of elements that commute with all other element ...
''c''
of the conformal field theory by
:
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 )
showed that the
Hausdorff dimension
In mathematics, Hausdorff dimension is a measure of ''roughness'', or more specifically, fractal dimension, that was first introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of ...
of the paths (with probability 1) is equal to min(2, 1 + ''κ''/8).
Left passage probability formulas for SLE''κ''
The probability of chordal SLE
''κ'' ''γ'' being on the left of fixed point
was computed by
:
where
is the
Gamma function
In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...
and
is the
hypergeometric function. This was derived by using the martingale property of
: