Related papers: Dimension and natural parametrization for SLE curv…
The natural paramterization or length for the Schramm-Loewner evolution (SLE{\kappa}) is the candidate for the scaling limit of the length of discrete curves for \kappa < 8. We improve the proof of the existence of the parametrization and…
The Schramm-Loewner evolution (SLE_\kappa) is a candidate for the scaling limit of random curves arising in two-dimensional critical phenomena. When \kappa < 8, an instance of SLE_\kappa is a random planar curve with almost sure Hausdorff…
The scaling limits of a variety of critical two-dimensional lattice models are equal to the Schramm-Loewner evolution (SLE) for a suitable value of the parameter kappa. These lattice models have a natural parametrization of their random…
This article is meant to serve as a guide to recent developments in the study of the scaling limit of critical models. These new developments were made possible through the definition of the Stochastic Loewner Evolution (SLE) by Oded…
Schramm-Loewner Evolutions (SLEs) describe a one-parameter family of growth processes in the plane that have particular conformal invariance properties. For instance, SLE can define simple random curves in a simply connected domain. In this…
Let $\gamma$ be the curve generating a Schramm--Loewner Evolution (SLE) process, with parameter $\kappa\geq0$. We prove that, with probability one, the Hausdorff dimension of $\gamma$ is equal to $\operatorname {Min}(2,1+\kappa/8)$.
In this paper, we will show that the higher moments of the natural parametrization of SLE curves in any bounded domain in the upper half plane is finite. We prove this by estimating the probability that an SLE curve gets near n given…
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 intuitive ideas that lead to the definition…
We use the interpretation of the Schramm-Loewner evolution as a limit of path measures tilted by a loop term in order to motivate the definition of $n$-radial SLE going to a particular point. In order to justify the definition we prove that…
In recent work we have shown that loop-erased random walk (LERW) connecting two boundary points of a domain converges to the chordal Schramm-Loewner evolution (SLE(2)) in the sense of curves parametrized by Minkowski content. In this note…
In the last few years, new insights have permitted unexpected progress in the study of fractal shapes in two dimensions. A new approach, called Schramm-Loewner evolution, or SLE, has arisen through analytic function theory and probability…
Many mathematical models of statistical physics in two dimensions are either known or conjectured to exhibit conformal invariance. Over the years, physicists proposed predictions of various exponents describing the behavior of these models.…
We find optimal (up to constant) bounds for the following measures for the regularity of the Schramm-Loewner evolution (SLE): variation regularity, modulus of continuity, and law of the iterated logarithm. For the latter two we consider the…
Developing the theory of two-sided radial and chordal $\mathit{SLE}$, we prove that the natural parametrization on $\mathit{SLE}_{\kappa}$ curves is well defined for all $\kappa<8$. Our proof uses a two-interior-point local martingale.
Studying SLE$_{\kappa}$ on $S^2$ provides a new and interesting perspective for the conformality of some 2-dimensional physical models. We prove the existence and some basic properties of the spherical Minkowski content of SLE$_{\kappa}$,…
The scaling limit of the two-dimensional self-avoiding walk (SAW) is believed to be given by the Schramm-Loewner evolution (SLE) with the parameter kappa equal to 8/3. The scaling limit of the SAW has a natural parameterization and SLE has…
We numerically show that the statistical properties of the shortest path on critical percolation clusters are consistent with the ones predicted for Schramm-Loewner evolution (SLE) curves for $\kappa=1.04\pm0.02$. The shortest path results…
Levy-Loewner evolution (LLE) is a generalization of the Schramm-Loewner evolution (SLE) where the branching is possible in a course of growth process. We consider a class of radial Levy-Loewner evolutions for which sets of points of the…
SLE is a random growth process based on Loewner's equation with driving parameter a one-dimensional Brownian motion running with speed $\kappa$. This process is intimately connected with scaling limits of percolation clusters and with the…
We consider shape, size and regularity of the hulls of the chordal Schramm-Loewner evolution driven by a symmetric alpha-stable process. We obtain derivative estimates, show that the complements of the hulls are Hoelder domains, prove that…