Related papers: Up-to-constants bounds on the two-point Green's fu…
Simmons and Cardy recently predicted a formula for the probability that the chordal SLE(8/3) path passes to the left of two points in the upper half-plane. In this paper we give a rigorous proof of their formula. Starting from this result,…
In this note, we prove a version of the conjectured duality of Schramm-Loewner Evolutions, by establishing exact identities in distribution between some boundary arcs of chordal $\SLE_\kappa$, $\kappa>4$, and appropriate versions of…
SLE$_{\kappa}(\rho)$ is a variant of SLE$_{\kappa}$ where $\rho$ characterizes the repulsion (if $\rho>0$) or attraction $(\rho<0)$ from the boundary. This paper examines the probabilities of SLE$_{\kappa}(\rho)$ to get close to the…
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…
In previous work [AHP24], we proved a finite-time large deviation principle in the Hausdorff metric for multiradial Schramm-Loewner evolution, SLE$(\kappa)$, as $\kappa \to 0$, with good rate function being the multiradial Loewner energy.…
We study the commutation relation for 2-radial SLE in the unit disc starting from two boundary points. We follow the framework introduced by Dub\'{e}dat. Under an additional requirement of the interchangeability of the two curves, we…
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…
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…
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…
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 improve the geometric properties of SLE$(\kappa;\vec{\rho})$ processes derived in an earlier paper, which are then used to obtain more results about the duality of SLE. We find that for $\kappa\in (4,8)$, the boundary of a standard…
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 $\operatorname {Min}(2,1+\kappa/8)$.
When studying stochastic processes, it is often fruitful to have an understanding of several different notions of regularity. One such notion is the optimal H\"older exponent obtainable under reparametrization. In this paper, we show that…
We show that in the continuum limit watersheds dividing drainage basins are Schramm-Loewner Evolution (SLE) curves, being described by one single parameter $\kappa$. Several numerical evaluations are applied to ascertain this. All…
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 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…
Various features of the two-parameter family of Schramm-Loewner evolutions SLE(\kappa,\rho) are studied. In particular, we derive certain restriction properties that lead to a ``strong duality'' conjecture, which is an identity in law…
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.
Standard stochastic Loewner evolution (SLE) is driven by a continuous Brownian motion, which then produces a continuous fractal trace. If jumps are added to the driving function, the trace branches. We consider a generalized SLE driven by a…
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…