Schramm-Löwner evolution

The Schramm-Löwner Evolution (SLE, also Stochastic Löwner Evolution, where English is mostly written as Loewner) from the field of stochastic geometry describes a one-parameter family of flat curves that are formed with a random law. They are conformally invariant, which creates connections to complex analysis, and enabled advances in the strict treatment of many models of statistical mechanics and their behavior at the critical point ( Ising model , percolation theory , self-avoiding paths , various random walk variants such as loop erased random walk, LERW), the so-called scaling limit case, which describes a phase transition in which the model becomes scale-invariant and thus conformally invariant. The curves are examples of fractals .

The SLE was introduced in 2000 by Oded Schramm , whereby the Loewner referred to a contribution from the function theory by Charles Loewner from 1923.

From the 2000s onwards, the SLE was one of the most active research areas in probability theory with two Fields Medals for research in this area ( Wendelin Werner , Stanislaw Smirnow ).

Story and motivation

Systems of statistical mechanics at the critical point (phase transition) have been described with great success since the end of the 1960s by the method of the renormalization group developed by Kenneth Wilson , which used methods originally derived from the quantum field theory of elementary particle physics (only in Euclidean spaces instead of in Minkowski space ). Another important step was the application of conformal field theories to two-dimensional systems of statistical mechanics. The SLE emerged from the endeavor of mathematicians to give the results and assumptions made by physicists a mathematically exact basis. In particular, they sought rigorous evidence for conjectures such as that of John Cardy that, in critical percolation, clusters should connect the edge points of simply connected areas in which the percolation model is being considered, and wanted to better understand the exact mathematical meaning of the successes of using conformal field theories in statistical physics understand. For this purpose, the “temporal” development (evolution) of a continuous curve that starts from the edge of a simply connected area in the plain was examined . This can serve as a model for the edge curve of a cluster in percolation theory (which are defined as a discrete model on a grid, but one can consider the continuum limit case of vanishing grid spacings). ${\displaystyle D}$ ${\displaystyle \gamma (t)}$ ${\displaystyle D}$

Löwner had examined a similar model for conformal mapping. He considered a curve that starts from an edge point of a simply connected area of the complex plane, the evolution being given by a real continuous function . The curve should not intersect itself. Loewner did not consider the behavior of the curve directly, but rather via its complement , which was simply connected and could therefore be mapped in conformity with one of the standard areas specified by Riemann's theorem of mapping (inside of the unit disk, upper half-plane , Riemann number sphere ). Löwner showed that this mapping is determined by the real function (the "driving function") and derived a differential equation named after him for the evolution of the curve with his construction (an evolution equation with an independent variable). ${\displaystyle \gamma (t)}$ ${\displaystyle D}$ ${\displaystyle \zeta (t)}$ ${\displaystyle D\backslash \gamma }$ ${\displaystyle \zeta (t)}$ ${\displaystyle t}$

In particular, the driving function could also describe a random process in a random curve . Schramm found that for conformal invariance of the associated probability measure on the curve, the random process of evolution had to be a one-dimensional Brownian motion , with a parameter in the form of a diffusion constant. ${\displaystyle \gamma }$ ${\displaystyle \zeta (t)}$ ${\displaystyle \kappa \geq 0}$

At Schramm, S in SLE stood for stochastic , later, in recognition of Schramm, it was made Schramm-Loewner evolution.

According to its assumptions, the theory should be applicable to all models of statistical mechanics in which a crossing prohibition exists for the corresponding curves and which are conformally invariant in the continuum limit case. In the case of grid models, the continuum limit case corresponds to the case that the grid spacing tends to zero. Conformal invariance is present at the critical point (phase transition) where the system looks similar across all length scales. Schramm, Wendelin Werner and Gregory Lawler showed in a series of papers that a whole series of properties at the critical point (such as various critical exponents, the Hausdorff dimension of various limiting curves) could be strictly mathematically derived with SLE (if the mentioned requirements were met ). When Stanislaw Smirnow was able to prove conformal invariance in the continuum limit case of percolation on the triangular grid, the critical exponents in the case of two-dimensional percolation were mathematically strictly derived. It is assumed that conformal invariance in the continuum limit case is also present in other important lattice models of statistical mechanics such as the Ising model, the XY model, and the Potts model. Each of these models corresponds to a specific choice of the parameter , the associated SLE is denoted by. ${\displaystyle O(N)}$ ${\displaystyle \kappa }$ ${\displaystyle SLE_{\kappa }}$

definition

The SLE is defined as a complex-valued solution of the Löwner differential equation with a driving real function that corresponds to a one-dimensional Brownian motion. A random curve is generated by the construction, but at the same time it is conformally invariant. It has one parameter . A distinction is made between radial SLE, chordal SLE and SLE in the entire complex plane. ${\displaystyle \zeta (t)}$ ${\displaystyle \kappa >0}$

A simply connected open area and a curve ( ) in . The endpoints are and . With chordal SLE the curve connects two edge points , with radial SLE one edge point with an inner point of , with SLE two points in the entire complex plane . Often the upper half-plane is mapped for the chordal SLE (with the edge points on the real axis and at infinity) and for the radial SLE the interior of the unit circle (with the SLE curve connecting an edge point with the origin). ${\displaystyle D\subseteq \mathbb {C} }$ ${\displaystyle \gamma (t)}$ ${\displaystyle t\geq 0}$ ${\displaystyle D}$ ${\displaystyle \gamma (0)=z}$ ${\displaystyle \gamma (t)=w}$ ${\displaystyle \partial D}$ ${\displaystyle D}$ ${\displaystyle \mathbb {C} }$

${\displaystyle D}$If, for example, in the case of the chordal SLE, the upper half-plane (including the point at infinity) , where has an initial value for on the real axis, then an analytical, one-to-one function is used to map conformally to (the independent variable t appears here as a subscript Index). is called the SLE curve or track of the SLE. be the inverse function. After suitable normalization, the Löwner differential equation applies to the evolution of the curve ${\displaystyle \mathbb {H} }$ ${\displaystyle \gamma (\zeta (t))}$ ${\displaystyle t=0}$ ${\displaystyle \mathbb {H} \backslash \gamma }$ ${\displaystyle f_{t}}$ ${\displaystyle \mathbb {H} }$ ${\displaystyle \gamma }$ ${\displaystyle g_{t}}$

${\displaystyle {\frac {\partial f_{t}(z)}{\partial t}}={\frac {df_{t}(z)}{dz}}{\frac {2}{\zeta (t)-z}}}$

${\displaystyle {\frac {\partial g_{t}(z)}{\partial t}}={\frac {2}{g_{t}(z)-\zeta (t)}}.}$

each for the functions . ${\displaystyle f_{t},g_{t}}$

If one chooses for the interior of the unit circle (radial SLE) one has ${\displaystyle D}$

${\displaystyle {\frac {\partial f_{t}(z)}{\partial t}}=-z{\frac {df_{t}(z)}{dz}}{\frac {\zeta (t)+z}{\zeta (t)-z}}}$

${\displaystyle {\dfrac {\partial g_{t}(z)}{\partial t}}=g_{t}(z){\dfrac {\zeta (t)+g_{t}(z)}{\zeta (t)-g_{t}(z)}}.}$

The analytical continuation of the mapping defined on or its reverse mapping between and : ${\displaystyle D\backslash \gamma }$ ${\displaystyle f_{t}}$ ${\displaystyle g_{t}}$ ${\displaystyle \zeta (t)}$ ${\displaystyle \gamma (t)}$

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

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

For the one-dimensional Brownian motion is taken with a diffusion constant . The resulting solution of the differential equation with initial value is called . ${\displaystyle \zeta (t)={\sqrt {\kappa }}B(t)}$ ${\displaystyle \kappa }$ ${\displaystyle g_{t}(z)}$ ${\displaystyle g_{0}(z)=z}$ ${\displaystyle SLE_{\kappa }}$

Special values ​​of the diffusion constant and results

• ${\displaystyle \kappa =2}$Loop Erased Random Walk (introduced by Greg Lawler), Schramm 2000 dealt with this case in his original work with the proof by Lawler, Schramm and Werner 2004. Closely related to it is the Uniform Spanning Tree (UST) (a random tree whose outline curve is connected with). ${\displaystyle SLE_{8}}$
• ${\displaystyle \kappa ={\frac {8}{3}}}$ Edge curves of the clusters with Brownian movement and presumably the scaling limit value of the Self Avoiding Random Walk
• ${\displaystyle \kappa =3}$ Edge curves of the clusters in the Ising model
• ${\displaystyle 0\leq \kappa \leq 4}$ the curve is simple (it doesn't cut itself)
• ${\displaystyle \kappa =4}$ Harmonic Explorer
• ${\displaystyle 4<\kappa <8}$ the curve intersects itself, every point is enclosed by the curve, but it does not fill space.
• for is the curve space-filling (with probability 1) ${\displaystyle \kappa \geq 8}$
• ${\displaystyle \kappa =6}$critical percolation on the triangular grid (Smirnov 2001) and probably also on other grids. This enabled the strict derivation of critical exponents with plane percolation (with earlier work by Harry Kesten ).

The conformal field theories corresponding to SLE, have central charge c with

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

For this corresponds to two values ​​of : one value is in , the other is also "dual" and greater than four. ${\displaystyle c<1}$ ${\displaystyle \kappa }$ ${\displaystyle 0\leq \kappa \leq 4}$ ${\displaystyle {\frac {16}{\kappa }}}$

According to Vincent Beffara , the Hausdorff dimension of the SLE curves is equal to the minimum of and (with probability 1). In other words, the fractal dimension increases from the value of a plane curve in the case (No stochastic motion) to the value from , the maximum possible value in one plane (dimension 2, space-filling curve). In 2001, Lawler, Schramm and Werner showed that the fractal dimension of the edge is plane Brownian motion . ${\displaystyle 2}$ ${\displaystyle 1+{\frac {\kappa }{8}}}$ ${\displaystyle 1}$ ${\displaystyle \kappa =0}$ ${\displaystyle 2}$ ${\displaystyle \kappa =8}$ ${\displaystyle SLE_{6}}$ ${\displaystyle {\frac {4}{3}}}$

literature

Original works:

• Oded Schramm: Scaling limits of loop-erased random walks and uniform spanning trees , Israel Journal of Mathematics, Volume 118, 2000, pp. 221-288, Arxiv
• Oded Schramm: Conformally invariant scaling limits: an overview and a collection of problems , ICM 2006
• Gregory Lawler, Oded Schramm, Wendelin Werner: Values ​​of Brownian intersection exponents, part 1-3, Acta Math., Volume 187, 2001, pp. 237-273, 275-308, Ann. Inst. Henri Poincaré (Statistique), Volume 38, 2002, pp. 109-123, Part 1, Arxiv , Part 2, Arxiv , Part 3, Arxiv
• Lawler, Schramm, Werner: Conformal invariance of planar loop-erased random walks and uniform spanning trees, Annals of Probability, Volume 32, 2004, pp. 939-995, Arxiv
• Schramm, Lawler, Werner: Conformal restriction: the chordal case, J. Am. Math. Soc., Vol. 16, 2003, pp. 917-955, Arxiv
• Steffen Rohde, Oded Schramm: Basic properties of SLE, Annals of Mathematics, Volume 161, 2005, pp. 879-920, Arxiv
• Stanislav Smirnov: Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits, CR Acad. Sci. Paris, Ser. Math. Volume 333, 2001, pp. 239-244.
• Stanislav Smirnov: Conformal Invariance of 2D lattice models, ICM 2006, Arxiv

Reviews and books:

• Gregory Lawler: Conformally invariant processes in the plane, Mathematical Surveys and Monographs 114, AMS, 2005
• Gregory Lawler: Schramm-Loewner evolution, Park City Lectures 2007, Arxiv
• Wendelin Werner: Random planar curves and Schramm-Loewner evolutions, in: Lectures on probability theory and statistics, Lecture Notes in Mathematics 1840, Springer 2004, pp. 107-19, Arxiv
• John Cardy: SLE for theoretical physicists, Annals of Physics, Volume 318, 2005, pp. 81-118, Arxiv
• Wouter Kager, Bernard Nienhuis: A guide to stochastic Loewner evolution and its applications, J. Stat. Phys., Vol. 115, 2004, pp. 1149-1229, Arxiv
• Vincent Beffara, Hugo Duminil-Copin, Planar percolation with a glimpse at Schramm-Loewner, La Pietra week in probability 2011, Arxiv

Web links

• N. Berestycki, J. Norris, Lectures on Schramm-Loewner Evolution, pdf
• Vincent Beffara, Schramm-Loewner evolution and other conformally invariant objects
• Vincent Beffara, Raconte moi ... le SLE
• Lawler, An introduction to stochastic Loewner evolution, pdf

Individual evidence

1. ^ Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, "Israel Journal of Mathematics, Volume 118, 2000, pp. 221-288
2. ↑ In the percolation model when a critical value is exceeded with which grid points are occupied
3. ↑ Löwner, Investigations on simple, conformal images of the unit circle, I, Mathematische Annalen, Volume 89, 1923, pp. 103–121, digitized
4. ^ John Cardy, Annals of Physics, Volume 318, 2005
5. ↑ Lawler, Schramm, Werner, Conformal invariance of planar loop-erased random walks and uniform spanning trees, Annals of Probability, Volume 32, 2004, pp. 939-995
6. ↑ Lawler, Schramm, Werner, On the scaling limit of planar self-avoiding walk, in: Fractal geometry and applications: a jubilee of Benoit Mandelbrot, Symp. Pure Math. 72, Part 2, AMS, 2004, Arxiv
7. ↑ Smirnov, Hugo Duminil-Copin , Hongler, Kemppainen, Chelkak , Convergence of Ising interfaces to Schramm's SLE curves, Arxiv 2013
8. ^ Schramm, Scott Sheffield , The harmonic explorer and its convergence to SLE (4), Annals of Probability, Volume 33, 2005, 2127-2148, Arxiv
9. ^ Rohde, Schramm, Basic properties of SLE, Ann. of Math., Vol. 161, 2005, p. 879
10. ↑ Smirnov, Werner: Critical exponents for two-dimensional percolation, Math. Res. Lett., Volume 8, 2001, pp. 729-744, Arxiv
11. ^ Kesten, Scaling relations for 2D percolation, Comm. Math. Phys., Vol. 109, 1987, pp. 109-156
12. ↑ Michel Bauer, Denis Bernard, growth processes and conformal field theories, Phys. Lett. B, Vol. 543, 2002, pp. 135-138, Arxiv ${\displaystyle SLE_{\kappa }}$
13. ^ Bauer, Bernard, Conformal field theories of Stochastic Loewner Evolutions, Comm. Math. Phys., Vol. 239, 2003, pp. 493-521, Arxiv
14. ^ Beffara, The dimension of the SLE curves, Annals of Probability, Volume 36, 2008, pp. 1421-1452, Arxiv
15. ↑ Lawler, Schramm, Werner, The dimension of the planar Brownian frontier is 4/3, Mathematical Research Letters, Volume 8, 2001, pp. 401-411