Related papers: Duality of Chordal SLE
We examine three--dimensional turbulent flows in the presence of solid-body rotation and helical forcing in the framework of stochastic Schramm-L\"owner evolution curves (SLE). The data stems from a run on a grid of $1536^3$ points, with…
Two-pointed quantum disks with a weight parameter $W > 0$ are a family of finite-area random surfaces that arise naturally in Liouville quantum gravity. In this paper we show that conformally welding two quantum disks according to their…
A space of local martingales of SLE type growth processes forms a representation of Virasoro algebra, but apart from a few simplest cases not much is known about this representation. The purpose of this article is to exhibit examples of…
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$_4$…
The probability that a point is to one side of a curve in Schramm-Loewner evolution (SLE) can be obtained alternatively using boundary conformal field theory (BCFT). We extend the BCFT approach to treat two curves, forming, for example, the…
We establish a large deviation principle for chordal SLE$_\kappa$ parametrized by capacity, as the parameter $\kappa \to 0+$, in the topology generated by uniform convergence on compact intervals of the positive real line. The rate function…
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…
We give a new proof of the reversibility of the Schramm Loewner evolution for $\kappa \leq 4$. The main ideas used in the proof are similar to those used in the original proof of this result, given by Zhan.
We make use of the fact that a two-sided whole-plane Schramm-Loewner evolution (SLE$_\kappa$) curve $\gamma$ for $\kappa\in(0,8)$ from $\infty$ to $\infty$ through $0$ may be parametrized by its $d$-dimensional Minkowski content, where…
This paper concerns a random walk on a planar graph and presents certain estimates concerning the harmonic measures for the walk in a grid domain which estimates are useful for showing the convergence of a LERW (loop-erased random walk) to…
The Green's function for the chordal Schramm-Loewner evolution $SLE_\kappa$ for $0 < \kappa < 8$, gives the normalized probability of getting near points. We give up-to-constant bounds for the two-point Green's function.
In this paper we consider the natural random walk on a planar graph and scale it by a small positive number $\delta$. Given a simply connected domain $D$ and its two boundary points $a$ and $b$, we start the scaled walk at a vertex of the…
Appreciation of Stochastic Loewner evolution (SLE$_\kappa$), as a powerful tool to check for conformal invariant properties of geometrical features of critical systems has been rising. In this paper we use this method to check conformal…
Schramm Loewner Evolutions (SLE) are random increasing hulls defined through the Loewner equation driven by Brownian motion. It is known that the increasing hulls are generated by continuous curves. When the driving process is of the form…
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…
Amorphous solids may resist external deformation such as shear or compression while they do not present any long-range translational order or symmetry at the microscopic scale. Yet, it was recently discovered that, when they become rigid,…
We show that the modulus of continuity of the SLE$_4$ uniformizing map is given by $(\log \delta^{-1})^{-1/3+o(1)}$ as $\delta \to 0$. As a consequence of our analysis, we show that the Jones-Smirnov condition for conformal removability…
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$_{\kappa}$,…
We characterize and describe all random subsets $K$ of a given simply connected planar domain (the upper half-plane $\H$, say) which satisfy the ``conformal restriction'' property, i.e., $K$ connects two fixed boundary points (0 and…
This article pertains to the classification of pairs of simple random curves with conformal Markov property and symmetry. We give the complete classification of such curves: conformal Markov property and symmetry single out a two-parameter…