Related papers: Non-simple SLE curves are not determined by their …

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 develop a percolation model for nodal domains in the eigenvectors of quantum chaotic torus maps. Our model follows directly from the assumption that the quantum maps are described by random matrix theory. Its accuracy in predicting…

We show that, for $\kappa\in(0,8)$, the integral of the laws of two-sided radial SLE$_\kappa$ curves through different interior points against a measure with SLE$_\kappa$ Green function density is the law of a chordal SLE$_\kappa$ curve,…

We derive a number of estimates for the probability that a chordal SLE path in the upper half plane H intersects a semicircle centred on the real line. We prove that if 0<kappa<8 and gamma:[0,infinity) to closure(H) is a chordal SLE in H…

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…

An annulus SLE$_\kappa$ trace tends to a single point on the target circle, and the density function of the end point satisfies some differential equation. Some martingales or local martingales are found for annulus SLE$_4$, SLE$_8$ and…

In this paper, we construct and then prove the up-to constants uniqueness of the natural measure on several random fractals, namely the SLE cut points, SLE boundary touching points, CLE pivotal points and the CLE carpet/gasket. As an…

We establish an upper bound on the asymptotic probability of an SLE(kappa) curve hitting two small intervals on the real line as the interval width goes to zero, for the range 4 < kappa < 8. As a consequence we are able to prove that 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.

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…

We show that, under mild assumptions on the limiting curve, a sequence of simple chordal planar curves converges uniformly whenever certain Loewner driving functions converge. We extend this result to random curves. The random version…

We present a review of the recent progress on percolation scaling limits in two dimensions. In particular, we will consider the convergence of critical crossing probabilities to Cardy's formula and of the critical exploration path to…

We study SLE$_{\kappa}$ theory with elements of Quasi-Sure Stochastic Analysis through Aggregation. Specifically, we show how the latter can be used to construct the SLE$_{\kappa}$ traces quasi-surely (i.e. simultaneously for a family of…

We give a simplified and complete proof of the convergence of the chordal exploration process in critical FK-Ising percolation to chordal SLE$_\kappa( \kappa-6)$ with $\kappa=16/3$. Our proof follows the classical excursion-construction of…

SLE(kappa,rho) is a generalisation of Schramm-Loewner evolution which describes planar curves which are statistically self-similar but not conformally invariant in the strict sense. We show that, in the context of boundary conformal field…

We prove that the interface of critical site percolation on the triangular lattice converges to SLE$_6$ in its natural parametrization, where the discrete interface is parametrized such that each edge is crossed in one unit of time, while…

We establish existence and uniqueness for Gaussian free field flow lines started at {\em interior} points of a planar domain. We interpret these as rays of a random geometry with imaginary curvature and describe the way distinct rays…

This is an introductory account of the emergence of conformal invariance in the scaling limit of planar critical percolation. We give an exposition of Smirnov's theorem (2001) on the conformal invariance of crossing probabilities in site…

This paper initiates the study of the conformal field theory of the SLE$_\kappa$ loop measure $\nu$ for $\kappa\in(0,4]$, the range where the loop is almost surely simple. First, we construct two commuting representations…

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…