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Computational Complexity pp 3075–3096Cite as

Stochastic Loewner Evolution: Linking Universality, Criticality and Conformal Invariance in Complex Systems

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Article Outline

Glossary

Definition of the Subject

Introduction

Scaling

Conformal Invariance

Loewner Evolution

Stochastic Loewner Evolution

Results and Discussion

Future Directions

Acknowledgments

Bibliography

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Abbreviations

Bessel process :

The Bessel process operates in d dimensions and describes the radial distance r from the origin of a particle performing Brownian motion. The random motion is governed by the Langevin equation \( { \text{d} r/\text{d} t=\kappa(d-1)/2r+\xi } \), where \( { \langle\xi\xi\rangle(t)=\kappa\delta(t) } \). For \( { d\leq 2 } \) the motion is recurrent, i. e., returns to the origin; for \( { d > 2 } \) the motion goes off to infinity. In SLE the Bessel process describes the transition between simple curves and self‐intersecting curves.

Brownian motion :

Brownian motion, B t , is the scaling limit of random walk. Brownian motion is plane‐filling, has the fractal dimension \( { D=2 } \), and is described by the Langevin equation \( \text{d} B_t/\text{d} t=\eta_t, \langle\eta_t\eta_s\rangle= {\delta(t-s)} \). B t is characterized by i) the stationarity property, \( B_{t+t^{\prime}}-B_{t} \) and \( B_{t^{\prime}} \) identical in distribution and ii) the independence property, \( B_{\Delta t^{\prime}} \) and \( B_{\Delta t} \) independent for \( \Delta t\neq\Delta t^{\prime} \). The correlations are given by \( \langle\vert B_t-B_{s}\vert ^2\rangle=\vert t-s\vert \) and B t is distributed according to the Gaussian (normal) distribution \( P(B,t)=(2\pi t)^{-1/2}\exp(-B^2/2t) \).

Chordal SLE :

In order to map the geometry of the growing hull to a real function by means of SLE one chooses a reference domain. Chordal SLE refers to the case where the SLE trace is grown between two boundary points, usually the origin and the point at infinity in the upper half complex plane.

Conformal invariance :

Conformal invariance or local scale invariance is a larger symmetry than scale invariance. Conformal invariance implies invariance under both a local rotation, translation, and dilatation. Conformal invariance is particularly powerful in 2D where a conformal transformation is implemented by an analytic function.

Conformal transformation :

A conformal transformation is a transformation which preserves angles but allows for rotation, dilatation, and translation. In an elastic medium picture a conformal transformation corresponds to translation, rotation, and compression (dilatation), but not shear. In 2D an analytic complex function \( { w=f(z) } \) from the complex z plane to the complex w plane generates a conformal transformation.

Continuum limit :

The limit where a lattice model approaches a continuum model; also called the scaling limit. The resulting continuum field theories define the universality classes of the lattice models.

Correlation length :

The correlation length measures the size of correlations. At the critical point the correlation length diverges signaling that the system becomes scale invariant.

Critical curves :

Critical curves are domain walls or cluster boundaries at the critical point. Critical curves are scale invariant and characterized by a fractal dimension.

Exploration :

Domain walls at the critical point can mathematically be generated by an exploration process where the domain wall is initiated at a boundary point and constructed in steps across the domain. In the percolation case the local step is generated by ‘flipping a coin’, in the Ising case by evaluating the magnetization at the tip of the ‘growing’ domain wall. The construction by an exploration process is essential in generating a critical curve by stochastic Loewner evolution.

Fortuin–Kasteleyn representation (FK):

Based on a high temperature expansion one can represent the configurations in the Ising, O(n), and Potts models by means of the Fortuin–Kasteleyn representation in terms of random clusters. The FK transformation is essential in identifying random curves for the lattice models which then can be accessed by SLE in the scaling limit.

Fractal dimension :

Irregular objects can be characterized by a fractal dimension. The fractal dimension D is derived by covering the object with \( { N(\ell) } \) intervals, disks, or spheres of linear dimension ℓ. By letting \( { \ell\rightarrow 0 } \) this definition resolves the fine scale fractal structure and the scaling relation \( { N(\ell)\propto (\ell)^{-D} } \), or \( { D=-\lim_{\ell\rightarrow 0}\ln(N(\ell)/\ln(\ell) } \), yields the fractal dimension. Examples of deterministic fractals are for example the Cantor set, the Koch curve, the Sierpinski gasket, and plane‐filling Hilbert and Peano curves. Random walk and diffusion limited aggregation (DLA) are examples of random physical fractals.

Hulls :

For a self‐intersecting SLE trace the trace and enclosed regions form a so‐called hull. Points in the hull cannot be connected to infinity without crossing the trace. The growing hull of a self‐intersecting SLE trace eventually exhausts the half plane.

Ising model :

The Ising model originates from the theory of magnetism and plays an important role in the theory of phase transitions and in general in statistical mechanics. The model is defined on a lattice where each lattice sites is endowed with a local spin variable assuming two distinct values. The spins interact by a short range exchange interaction. Above 1D the Ising model has a second order phase transition.

Locality :

In the percolation case the domain wall is constructed by an exploration process and a local rule for assigning the next step. This locality property is specific to percolation which has a geometric phase transition. In the SLE context the locality property implies \( { \kappa = 6 } \), i. e., the percolation case.

Loewner equation :

The Loewner equation is the first order nonlinear differential equation for the uniformizing map \( { g_t(z) } \) which maps the complement of a growing hull or curve in the upper half plane back to the upper half plane. The geometrical properties of the hull is encoded in the real function a t . The Loewner equation has the form \( { \text{d} g_t/\text{d} t=2/(g_t-a_t) } \).

Loewner evolution :

Loewner evolution refers to the parametrized map \( { g_t(z) } \) which uniformizes the complement of a curve or hull, say in the upper half plane.

Loop erased random walk (LERW):

Loop erased random walk is a variant of random walk where loops formed are removed as the walk progresses. LERW has a conformally invariant scaling limit and can be generated by SLE for \( { \kappa=2 } \).

Markov property :

The Markov property refers to a stochastic process without memory. A typical example is random walk or in the scaling limit Brownian motion. The Markov property is essential for the application of SLE to domain walls in the scaling limit.

Measures :

For lattice systems one can define probability distributions according to the rules of statistical mechanics. In the scaling limit the distributions typically diverge and have to be replaced by the more abstract mathematical concept of probability measures.

O(n) models :

The O(n) model is a generalization of the Ising model. At each lattice site is associated an n‑component spin variable. For \( { n=1 } \) we recover the Ising model, \( { n=2 } \) corresponds to the XY model, and \( { n=3 } \) to the Heisenberg model.

Peano curve :

A Peano curve is a non‐crossing curve which is dense in the plane, i. e., it gets arbitrarily close to every point. The Peano curve has the fractal dimension \( { D=2 } \). In a SLE context a random Peano curve winds around a UST and corresponds to \( { \kappa=8 } \).

Percolation :

Percolation on a lattice is constructed by filling lattice sites at random with a common probability p. At a critical concentration a spanning cluster extends across the system.

Potts model :

The Potts model is a generalization of the Ising model where the local lattice variable can assume q different values and where sites only interact when they are in the same Potts state. The Potts model has a FK cluster representation. The Ising model corresponds to \( { q=2 } \); \( { q=1 } \) corresponds to percolation and \( { q=0 } \) to the UST.

Random walk :

Random walk is an ubiquitous phenomenon in nature. In random walk on a square lattice a particle jump from site to site with a given fixed probability. Each step is independent of the past history, there is no memory, this is the so‐called Markov property. The random walk path or history is plane‐filling and has the fractal dimension \( { D=2 } \). The mean square displacement of random walk scales with the number of steps.

Restriction :

The restriction property in a SLE context implies that the measure on a curve in a domain D conditioned not to hit a bulge L is the same as the measure in the domain \( { D\setminus L } \). The restriction property holds in the case of a uniform measure and applies to self‐avoiding random walk.

Riemann's mapping theorem :

Riemann's mapping theorem states that an arbitrary simply connected domain D, i. e., without holes, can be mapped to another simply connected domain \( { D^{\prime} } \) by means of a suitable complex function g(z), i. e., \( { g(D)=D^{\prime} } \). Often the disk or half plane are used as reference domains. The mapping theorem does not make assumptions about the domain boundaries which can be fractal.

Scale invariance :

The scaling property refers to the case where a phenomenon is devoid of a characteristic scale or unit. Scale invariance is typically characterized by a power law behavior and critical exponents.

Scaling limit :

The limit where the lattice parameter in a lattice model approaches zero. The scaling limit is equivalent to the continuum limit.

Self‐avoiding random walk (SAW):

A self‐avoiding random walk is a walk conditioned not to cross itself. SAW has been used to model polymers. SAW has a uniform probability measure and conforms in a SLE context to the restriction condition.

Stochastic Loewner evolution (SLE):

Stochastic Loewner evolution is Loewner evolution driven by a real stochastic function a t with a distribution given by 1D Brownian motion, i. e., \( { a_t=\sqrt\kappa B_t } \). SLE is governed by the stochastic equation of motion \( { \text{d} g_t/\text{d} t=2/(g_t-a_t) } \).

Schramm's theorem :

Schramm's theorem refers to Schramm's derivation of SLE for LERW. Schramm showed that the scaling limit of LERW is described by SLE for \( { \kappa=2 } \). Schramm also conjectured that the random Peano curve winding around an UST is described by SLE for \( { \kappa=8 } \) and that percolation is described by SLE for \( { \kappa=6 } \).

Uniformizing maps :

Conformal transformations which map a domain D to a standard reference domain, e. g., the half plane or the disk, are called uniformizing maps.

Uniform spanning tree (UST):

A spanning tree is a collection of vertices and links forming a tree (no loops or cycles). A USF is a random tree picked among all spanning trees with equal probability.

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Acknowledgments

The present work has been supported by the Danish Natural Science Research Council.

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  1. Department of Physics and Astronomy, University of Aarhus, Aarhus, Denmark

    Hans C. Fogedby

  2. Niels Bohr Institute, Copenhagen, Denmark

    Hans C. Fogedby

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Fogedby, H.C. (2012). Stochastic Loewner Evolution: Linking Universality, Criticality and Conformal Invariance in Complex Systems. In: Meyers, R. (eds) Computational Complexity. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-1800-9_192

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