Related papers: Minkowski content and natural parameterization for…

Schramm-Loewner evolution (SLE$_\kappa$) is classically studied via Loewner evolution with half-plane capacity parametrization, driven by $\sqrt{\kappa}$ times Brownian motion. This yields a (half-plane) valued random field $\gamma = \gamma…

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…

It is known that Schramm-Loewner Evolutions (SLEs) have a.s. frontier points if $\kappa>4$ and a.s. cutpoints if $4<\kappa<8$. If $\kappa>4$, an appropriate version of $\SLE(\kappa)$ has a renewal property: it starts afresh after visiting…

We show that, for $\kappa\le 4$, the integral of the two-sided radial SLE$_\kappa$ measures over all interior points is chordal SLE$_\kappa$ biased by the path's natural length, which is its $(1+\kappa/8)$-dimensional Minkowski content.

SLE($\kappa,\rho$) is a variant of the Schramm-Loewner Evolution which describes the curves which are not conformal invariant, but are self-similar due to the presence of some other preferred points on the boundary. In this paper we study…

We prove that, for $\kappa\le 4$, backward chordal SLE$_\kappa$ admits backward chordal SLE$_\kappa(-4,-4)$ decomposition for the capacity parametrization. This means that, for any bounded measurable subset $U\subset Q_4:={\mathbb…

We use Minkowski content (i.e., natural parametrization) of SLE to construct several types of SLE$_\kappa$ loop measures for $\kappa\in(0,8)$. First, we construct rooted SLE$_\kappa$ loop measures in the Riemann sphere $\widehat{\mathbb…

We define multiple-paths Schramm-Loewner evolution ($SLE_\kappa$) in multiply connected domains when $\kappa\leq 4$ and prove that in annuli, the partition function is smooth. Moreover, we give up-to-constant estimates for the partition…

We prove upper bounds for the probability that a radial SLE$_{\kappa}$ curve, $\kappa\in(0,8)$, comes within specified radii of $n$ different points in the unit disc. Using this estimate, we then prove a similar upper bound for a…

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 construct the natural parametrisation of SLE$_4$ using the Gaussian free field, complementing the corresponding results for SLE$_\kappa$ for $\kappa \in (0,4)$ by Benoist and for $\kappa \in (4,8)$ by Miller and Schoug.

In the paper we prove that, for $\kappa\in(0,8)$, the $n$-point boundary Green's function of exponent $\frac8\kappa -1$ for chordal SLE$_\kappa$ exists. We also prove that the convergence is uniform over compact sets and the Green's…

We construct a natural measure mu supported on the intersection of a chordal SLE(kappa) curve gamma with the real line R, in the range 4 < kappa < 8. The measure is a function of the SLE path in question. Assuming that boundary measures…

We prove the existence and uniqueness of multiple SLE$_\kappa$ associated with any given link pattern for $\kappa\in (4,6]$. We also have the uniqueness for $\kappa\in (6,8)$. The multiple SLE$_\kappa$ law is constructed by first…

Stochastic Loewner Evolution (SLE_kappa) has been introduced as a description of the continuum limit of cluster boundaries in two-dimensional critical systems. We show that the problem of N radial SLEs in the unit disc is equivalent to…

The scaling limit of planar loop-erased random walks is described by a stochastic Loewner evolution with parameter kappa=2. In this note SLE(2) in the upper half-plane H minus a simply-connected compact subset K of H is studied. As a main…

We present an elementary proof establishing the equality of the right and left-sided $\sqrt{\kappa}$-quantum lengths for an SLE$_\kappa$ curve, where $\kappa\in (0,4]$. We achieve this by demonstrating that the$\sqrt{\kappa}$-quantum length…

We consider the Schramm-Loewner evolution (SLE$_\kappa$) with $\kappa=4$, the critical value of $\kappa > 0$ at or below which SLE$_\kappa$ is a simple curve and above which it is self-intersecting. We show that the range of an SLE$_4$…

This article employs Schramm-Loewner Evolution to obtain intersection exponents for several chordal $SLE_{8/3}$ curves in a wedge. As $SLE_{8/3}$ is believed to describe the continuum limit of self-avoiding walks, these exponents correspond…

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…