Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0706.1177

Stochastic Loewner Evolution

Hans C. Fogedby
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 method and demonstrated that driving the Loewner evolution by a one-dimensional Brownian motion, the curves in the complex plane become scale invariant; the fractal dimension turns out to be determined by the strength of the Brownian motion. SLE fills a gap in our understanding of the critical properties of a variety of lattice models in their scaling limits and supplements the result obtained by means of conformal field theory. In this paper we attempt to provide a simple and heuristic discussion of some of the important aspects of SLE.

Explore related subjects

cond-mat.stat-mech

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Hans C. Fogedby. 2007-06-08. Stochastic Loewner Evolution. https://arxiv.org/abs/0706.1177

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

First Passage Times for Variable-Order Time-Fractional Diffusion

We derive the asymptotic first passage time (FPT) distribution for space-dependent variable-order time-fractional diffusion, where the fractional exponent $α(x)$ varies with position. On a bounded interval with an absorbing and a reflecting boundary, we show that the survival probability decays as $Ψ(t)\sim C\,t^{-α_*}/[\ln (t/τ)]^ν$, where $α_*$ is the minimum value of the fractional exponent and $ν$ is determined by the location and shape of the minimum. The exponent $ν$ is fixed by the location and order of the minimum, taking the value $1/k$ at a $k$\textsuperscript{th}-order interior minimum, $1$ at the reflecting boundary and $2$ at the absorbing boundary. For a constant fractional exponent, $ν=0$, so the logarithmic factor distinguishes a continuously varying order field from a homogeneous medium. We recover both exponents from simulated first passage times by a direct linear fit, with no amplitude or geometric input. We validate the theory against exact Laplace-space solutions and Monte Carlo simulations for linear and nonlinear profiles of $α(x)$.

cond-mat.stat-mech↗
arXiv2026

The unique, universal entropy for complex systems

A unique, universal definition for the entropy of nonlinear, complex systems is proposed. The Shannon entropy is shown to measure energetic, configurational, and nonlinear degrees of freedom (DOF). An entropy calibrated to the asymptotic nonlinearity of the distribution reduces the nonlinear DOF to zero and modifies the configurational DOF with a generalized logarithm. Physical axioms regarding the asymptotic growth of states, the calibration from an informational scale, and nonlinear dynamics provide a foundation for the proof that the calibrated entropy is unique and comprehensive across the short- and long-range dependency classes. The informational scale quantifies the linear uncertainty independent of the nonlinearity. This entropy, when constrained by the informational scale, is maximized by the coupled stretched exponential distributions (CSED). A consistent thermodynamics for nonequilibrium systems is proposed with the temperature equal to the scale constraint divided by the Boltzmann constant. Physical and informational applications are highlighted.

cond-mat.stat-mech↗
arXiv2026

Exact amplitude relations for diffusion-limited aggregation

It has been known for several decades that the third moment of the multifractal spectrum of the harmonic measure for diffusion-limited aggregates is linked to the underlying fractal dimension of the cluster. We demonstrate, using an argument based on the Hastings-Levitov formulation of diffusion-limited aggregation (DLA) in two dimensions, an even stronger link, connecting the universal amplitude of the third moment to the cluster fractal dimension. This argument can be used for both the standard circular DLA as well as DLA in a cylinder (i.e., with periodic boundary conditions); in the latter case the relationship of the amplitude to the fractal dimension is weaker, and must be extracted via a scaling analysis.

cond-mat.stat-mech↗