Restricting SLE(8/3) to an annulus

We study the probability that chordal $\text{SLE}_{8/3}$ in the unit disk from $\exp(ix)$ to 1 avoids the disk of radius $q$ centered at zero. We find the initial/boundary-value problem satisfied by this probability as a function of $x$ and $a=\ln q$, and show that asymptotically as $q$ tends to one this probability decays like $\exp(-cx/(1-q))$ with $c=5\pi/8$ for $x\in[0,\pi]$. We also give a representation of this probability as a functional of a Legendre process.Comment: 28 pages, corrected proof of asymptotic dependenc

The Correlator Toolbox, Metrics and Moduli

We discuss the possible set of operators from various boundary conformal field theories to build meaningful correlators that lead via a Loewner type procedure to generalisations of SLE($\kappa,\rho$). We also highlight the necessity of moduli for a consistent kinematic description of these more general stochastic processes. As an illustration we give a geometric derivation of $\text{SLE}(\kappa,\rho)$ in terms of conformally invariant random growing compact subsets of polygons. The parameters $\rho_j$ are related to the exterior angles of the polygons. We also show that $\text{SLE}(\kappa,\rho)$ can be generated by a Brownian motion in a gravitational background, where the metric and the Brownian motion are coupled. The metric is obtained as the pull-back of the Euclidean metric of a fluctuating polygon.Comment: 3 figure

Stochastic Loewner evolution in multiply connected domains

We construct radial stochastic Loewner evolution in multiply connected domains, choosing the unit disk with concentric circular slits as a family of standard domains. The natural driving function or input is a diffusion on the associated Teichm\"uller space. The diffusion stops when it reaches the boundary of the Teichm\"uller space. We show that for this driving function the family of random growing compacts has a phase transition for $\kappa=4$ and $\kappa=8$, and that it satisfies locality for $\kappa=6$.Comment: Corrected version, references adde

Chordal Loewner families and univalent Cauchy transforms

AbstractWe study chordal Loewner families in the upper half-plane and show that they have a parametric representation. We show one, that to every chordal Loewner family there corresponds a unique measurable family of probability measures on the real line, and two, that to every measurable family of probability measures on the real line there corresponds a unique chordal Loewner family. In both cases the correspondence is being given by solving the chordal Loewner equation. We use this to show that any probability measure on the real line with finite variance and mean zero has univalent Cauchy transform if and only if it belongs to some chordal Loewner family. If the probability measure has compact support we give two further necessary and sufficient conditions for the univalence of the Cauchy transform, the first in terms of the transfinite diameter of the complement of the image domain of the reciprocal Cauchy transform, and the second in terms of moment inequalities corresponding to the Grunsky inequalities