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

@inproceedings{Fogedby2007StochasticLE, title={Stochastic Loewner Evolution: Linking Universality, Criticality and Conformal Invariance in Complex Systems}, author={Hans C. Fogedby}, booktitle={Encyclopedia of Complexity and Systems Science}, year={2007} }

Stochastic Loewner evolution also called Schramm Loewner evolution (abbreviated, SLE) is a rigorous tool in mathematics and statistical physics for generating and studying scale invariant or fractal random curves in two dimensions. The method is based on the older deterministic Loewner evolution introduced by Karl Loewner, who demonstrated that an arbitrary curve not crossing itself can be generated by a real function by means of a conformal transformation. In 2000 Oded Schramm extended this…

## One Citation

### Stochastic games

- EconomicsProceedings of the National Academy of Sciences
- 2015

The historical context and the impact of Shapley’s contribution to stochastic games, which were the first general dynamic model of a game to be defined, are summarized.

## References

SHOWING 1-10 OF 101 REFERENCES

### Conformally Invariant Processes in the Plane

- Mathematics, Physics
- 2005

Theoretical physicists have predicted that the scaling limits of many two-dimensional lattice models in statistical physics are in some sense conformally invariant. This belief has allowed physicists…

### Conformal Field Theories of Stochastic Loewner Evolutions

- Mathematics
- 2003

Stochastic Loewner evolutions (SLEκ) are random growth processes of sets, called hulls, embedded in the two dimensional upper half plane. We elaborate and develop a relation between SLEκ evolutions…

### The dimension of the SLE curves

- Mathematics
- 2008

Let γ be the curve generating a Schramm–Loewner Evolution (SLE) process, with parameter κ ≥ 0. We prove that, with probability one, the Haus-dorff dimension of γ is equal to Min(2, 1 + κ/8).…

### Conformal Invariance and Stochastic Loewner Evolution Predictions for the 2D Self-Avoiding Walk—Monte Carlo Tests

- Mathematics
- 2004

AbstractSimulations of the two-dimensional self-avoiding walk (SAW) are performed in a half-plane and a cut-plane (the complex plane with the positive real axis removed) using the pivot algorithm. We…

### Conformal invariance and stochastic Loewner evolution processes in two-dimensional Ising spin glasses.

- MathematicsPhysical review letters
- 2006

We present numerical evidence that the techniques of conformal field theory might be applicable to two-dimensional Ising spin glasses with Gaussian bond distributions. It is shown that certain domain…

### Stochastic Loewner evolution driven by Lévy processes

- Mathematics
- 2006

Standard stochastic Loewner evolution (SLE) is driven by a continuous Brownian motion, which then produces a continuous fractal trace. If jumps are added to the driving function, the trace branches.…

### Random planar curves and Schramm-Loewner evolutions

- Geology
- 2003

We review some of the results that have been derived in the last years on conformal invariance, scaling limits and properties of some two-dimensional random curves. In particular, we describe the…

### Conformal Transformations and the SLE Partition Function Martingale

- Mathematics
- 2004

Abstract.
We present an implementation in conformal field theory (CFT) of local
finite conformal transformations fixing a point. We give explicit constructions when
the fixed point is either the…

### A Guide to Stochastic Löwner Evolution and Its Applications

- Mathematics
- 2003

This article is meant to serve as a guide to recent developments in the study of the scaling limit of critical models made possible through the definition of the Stochastic Löwner Evolution (SLE), and defines SLE and discusses some of its properties.