Schramm–Loewner evolution

In probability theory, Schramm–Loewner evolution, also known as stochastic Loewner evolution or SLE, is a conformally invariant stochastic process. It is a family of random planar curves that are generated by solving Charles Loewner's differential equation with Brownian motion as input. It was discovered by harvs|txt|first=Oded |last=Schramm|authorlink=Oded Schramm|year=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. Schramm–Loewner evolution is conjectured or proved to be the scaling limit of various critical percolation models, and other stochastic processes in the plane.

The Loewner equation

If "D" is a simply connected, open 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 and γ((0,∞)) in "D"), then for each "t"≥0, the complement "D"_"t" of γ( [0,"t"] ) is simply connected and therefore conformally isomorphic to "D" by the Riemann mapping theorem. If "f"_"t" is a suitable normalized isomorphism from "D" to "D"_"t", then it satisfies a differential equation found by harvtxt|Loewner|1923|loc=p. 121 in his work on the Bieberbach conjecture.Sometimes it is more convenient to use the inverse function "g"_"t" of "f"_"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 "f"_"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:$frac\{partial\; f\_t(z)\}\{partial\; t\}\; =\; -z\; f^prime\_t(z)frac\{zeta(t)+z\}\{zeta(t)-z\}$   or   $frac\{partial\; g\_t(z)\}\{partial\; t\}\; =\; g\_t(z)frac\{zeta(t)+g\_t(z)\}\{zeta(t)-g\_t(z)\}.$

When "D" is the upper half plane the Loewner equation differs from this by changes of variable and is :$frac\{partial\; f\_t(z)\}\{partial\; t\}\; =\; frac\{\; 2f\_t^prime(z)\}\{zeta(t)-z\}$   or   $frac\{partial\; g\_t(z)\}\{partial\; t\}\; =\; frac\{\; 2\}\{g\_t(z)-zeta(t)\}.$

The driving function ζ and the curve γ are related by :$displaystyle\; f\_t(zeta(t))\; =\; gamma(t)$   or   $displaystyle\; zeta(t)\; =\; g\_t(gamma(t))$where "f"_"t" and "g"_"t" are extended by continuity.

Example

If "D" is the upper half plane and the driving function ζ is identically zero, then :$f\_t(z)\; =\; sqrt\{z^2-4t\}$:$g\_t(z)\; =\; sqrt\{z^2+4t\}$:$gamma(t)\; =\; 2isqrt\{t\}$:$D\_t$ is the upper half plane with the line from 0 to $2isqrt\{t\}$ removed.

chramm-Loewner evolution

Schramm-Loewner evolution is the random curve γ given by the Loewner equation as in the previous section, for the driving function :$displaystylezeta(t)=B(kappa\; t)$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 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.

pecial values of κ

*κ=2 corresponds to the loop-erased random walk and the uniform spanning tree
*For κ=8/3 SLE_κ has the restriction property and is the scaling limit of self-avoiding walks. This case also arises in the scaling limit of critical percolation on the triangular lattice.
*For 0≤κ≤4 the curve γ("t") is simple (with probability 1)
*κ=4 corresponds to the harmonic explorer
*For κ=6 SLE_κ has the locality property. This arises in the scaling limit of critical percolation on the triangular lattice.
*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)

When SLE corresponds to some conformal field theory, the parameter &kappa; is related to the central charge "c"of the conformal field theory by:$c\; =\; frac\{(8-3kappa)(kappa-6)\}\{2kappa\}.$Each value of "c"<1 corresponds to two values of &kappa;, one value &kappa; between 0 and 4, and a "dual" value 16/&kappa; greater than 4.

Applications

harvtxt|Lawler|Schramm|Werner|2001 used SLE₆ to prove the conjecture of harvtxt|Mandelbrot|1982 that the boundary of planar Brownian motion has fractal dimension 4/3.

References

*citation|first=John|last=Cardy|url=http://arxiv.org/abs/cond-mat/0503313|title=SLE for theoretical physicists|journal= Annals Phys. |volume=318 |year=2005|pages= 81-118
*springer|id=l/l060960|title=Löwner method|first=E.G.|last= Goluzina
*springer|id=L/l060950|title=Löwner equation|first=V.Ya. |last=Gutlyanskii
*citation|url=http://arxiv.org/abs/math-ph/0312056|title=A Guide to Stochastic Loewner Evolution and its Applications
first=Wouter |last=Kager|first2= Bernard|last2= Nienhuis|journal= J. Stat. Phys.|volume= 115|pages=1149-1229 |year=2004|doi= 10.1023/B:JOSS.0000028058.87266.be
*Citation | last1=Lawler | first1=Gregory F. | title=Random walks and geometry | url=http://www.math.duke.edu/~jose/esi.html | publisher=Walter de Gruyter GmbH & Co. KG, Berlin | id=MathSciNet | id = 2087784 | year=2004 | chapter=An introduction to the stochastic Loewner evolution | pages=261–293|editor1-first=Vadim A.|editor1-last= Kaimanovich | isbn=3-11-017237-2
*Citation | last1=Lawler | first1=Gregory F. | title=Conformally invariant processes in the plane | publisher=American Mathematical Society | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-3677-4 | id=MathSciNet | id = 2129588 | year=2005 | volume=114| url =http://books.google.com/books/p/ams?id=JHMzab3u6U8C
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*Citation | last1=Lawler | first1=Gregory F. |authorlink1=Gregory Lawler| last2=Schramm | first2=Oded | authorlink2=Oded Schramm| last3=Werner | first3=Wendelin | authorlink3=Wendelin Werner|title=The dimension of the planar Brownian frontier is 4/3 | url=http://www.mrlonline.org/mrl/2001-008-004/2001-008-004-001.html | id=MathSciNet | id = 1849257 | year=2001 | journal=Mathematical Research Letters | issn=1073-2780 | volume=8 | issue=4 | pages=401–411
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*Citation | last1=Mandelbrot | first1=Benoît | author1-link=Benoît Mandelbrot | title=The Fractal Geometry of Nature | publisher=W. H. Freeman | isbn=978-0-7167-1186-5 | year=1982
*citation|title=Introduction to stochastic Loewner evolutions|url=http://www.statslab.cam.ac.uk/~james/Lectures/sle.pdf|first=J. R. |last= Norris|year=2007
*Citation | last1=Schramm | first1=Oded | authorlink=Oded Schramm| title=Scaling limits of loop-erased random walks and uniform spanning trees | url=http://arxiv.org/abs/math.PR/9904022 | id=MathSciNet | id = 1776084 | year=2000 | journal=Israel Journal of Mathematics | issn=0021-2172 | volume=118 | pages=221–288 Schramm's original paper, introducing SLE
*Citation | last1=Schramm | first1=Oded | authorlink=Oded Schramm | title=International Congress of Mathematicians. Vol. I | publisher=Eur. Math. Soc., Zürich | id=MathSciNet | id = 2334202 | year=2007 | chapter=Conformally invariant scaling limits: an overview and a collection of problems | pages=513–543|url=http://arxiv.org/abs/math/0602151|ISBN= 978-3-03719-022-7
*Citation | last1=Werner | first1=Wendelin | authorlink=Wendelin Werner | title=Lectures on probability theory and statistics | url=http://arxiv.org/abs/math.PR/0303354 | publisher=Springer-Verlag | location=Berlin, New York | series=Lecture Notes in Math. | id=MathSciNet | id = 2079672 | year=2004 | volume=1840 | chapter=Random planar curves and Schramm-Loewner evolutions | pages=107–195|doi=10.1007/b96719|isbn= 978-3-540-21316-1
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External links

* [http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=144460 Schramm–Loewner evolution on arxiv.org]
*citation|url=http://www.msri.org/publications/ln/msri/2001/percolation/schramm/2/index.html |title=Tutorial: SLE |last2= Schramm|last1= Lawler|last3= Werner |year=2001
place= Lawrence Hall of Science, University of California, Berkeley ( video of MSRI lecture)
*citation|first=Oded |last=Schramm|url=http://www.msri.org/publications/ln/msri/2001/percolation/schramm/1/index.html |title=Conformally Invariant Scaling Limits and SLE|publisher=MSRI |year=2001 (Slides from a talk.)