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

  title={Stochastic Loewner Evolution: Linking Universality, Criticality and Conformal Invariance in Complex Systems},
  author={Hans C. Fogedby},
  booktitle={Encyclopedia of Complexity and Systems Science},
  • H. Fogedby
  • Published in
    Encyclopedia of Complexity…
    8 June 2007
  • Mathematics
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… 
1 Citations

Stochastic games

  • Eilon Solan
  • Economics
    Proceedings 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.



Conformally Invariant Processes in the Plane

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

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

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

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.

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

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

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

2D growth processes: SLE and Loewner chains

Conformal Transformations and the SLE Partition Function Martingale

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

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.