Related papers: Imaginary Geometry I: Interacting SLEs

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…

Given a simply connected planar domain D, distinct points x,y \in \partial D, and \kappa >0, the Schramm-Loewner evolution SLE_\kappa is a random continuous non-self-crossing path in the closure of D from x to y. The…

We derive a surprising correspondence between SLE$_{\kappa}(\rho)$ processes and light cones of the Gaussian free field (GFF). Recall that (one-sided, chordal, origin-seeded) SLE$_\kappa(\rho)$ processes are in some sense the simplest and…

We compute the almost-sure Hausdorff dimension of the double points of chordal SLE_kappa for kappa > 4, confirming a prediction of Duplantier-Saleur (1989) for the contours of the FK model. We also compute the dimension of the cut points of…

Suppose that D is a planar Jordan domain and x and y are distinct boundary points of D. Fix \kappa \in (4,8) and let \eta\ be an SLE_\kappa process from x to y in D. We prove that the law of the time-reversal of \eta is, up to…

We study SLE$_\kappa(\rho)$ curves, with $\kappa$ and $\rho$ chosen so that the curves hit the boundary. More precisely, we study the sets on which the curves collide with the boundary at a prescribed "angle" and determine the almost sure…

Suppose that $h$ is a Gaussian free field (GFF) on a planar domain. Fix $\kappa \in (0,4)$. The SLE$_\kappa$ light cone ${\mathbf L}(\theta)$ of $h$ with opening angle $\theta \in [0,\pi]$ is the set of points reachable from a given…

We construct a conformally invariant random family of closed curves in the plane by welding of random homeomorphisms of the unit circle given in terms of the exponential of Gaussian Free Field. We conjecture that our curves are locally…

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 consider chordal SLE(kappa) curves for kappa > 4, where the intersection of the curve with the boundary is a random fractal of almost sure Hausdorff dimension min {2-8/kappa,1}. We study the random sets of points at which the curve…

We consider the motion by curvature of a network of smooth curves with multiple junctions in the plane, that is, the geometric gradient flow associated to the length functional. Such a flow represents the evolution of a two--dimensional…

Questions regarding the continuity in $\kappa$ of the $SLE_{\kappa}$ traces and maps appear very naturally in the study of SLE. In order to study the first question, we consider a natural coupling of SLE traces: for different values of…

A coarse grained description of a two phase fluid is used to study the steady state configuration of the interface separating the coexisting phases, and the motion of the contact line at which the interface intersects a solid boundary. The…

A moving contact line occurs at the intersection of an interface formed between two immiscible liquids and a solid. According to viscous theory, the flow is entirely governed by just two parameters, the viscosity ratio, $\lambda$, and the…

Conformal loop ensembles are random collections of loops in a simply connected domain, whose laws are characterized by a natural conformal invariance property. The set of points not surrounded by any CLE loop is a natural random and…

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…

Let $h$ be an instance of Gaussian Free Field in a planar domain. We study level lines of $h$ starting from boundary points. We show that the level lines are random continuous curves which are variants of SLE$_4$ path. We show that the…

The mean ($\kappa$) and Gaussian ($\bar{\kappa}$) bending rigidities of liquid-liquid interfaces, of importance for shape fluctuations and topology of interfaces, respectively, are not yet established: even their signs are debated. Using…

Let f:\Sigma_1 --> \Sigma_2 be a map between compact Riemannian manifolds of constant curvature. This article considers the evolution of the graph of f in the product of \Sigma_1 and \Sigma_2 by the mean curvature flow. Under suitable…

Random advection of Lagrangian tracer scalar field $\theta (t,x)$ by a one-dimensional, spatially smooth and short-correlated in time velocity field is considered. Scalar fluctuations are maintained by a source concentrated at the integral…