English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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 agiven boundary point by angle-varying flow lines of the (formal) vector field$e^{i h/\chi}$, $\chi = \tfrac{2}{\sqrt{\kappa}} - \tfrac{\sqrt{\kappa}}{2}$,with angles in $[-\tfrac{\theta}{2},\tfrac{\theta}{2}]$. We derive theHausdorff dimension of ${\mathbf L}(\theta)$.  If $\theta =0$ then ${\mathbf L}(\theta)$ is an ordinary SLE$_{\kappa}$ curve(with $\kappa < 4$); if $\theta = \pi$ then ${\mathbf L}(\theta)$ is the rangeof an SLE$_{\kappa'}$ curve ($\kappa' = 16/\kappa > 4$). In these extremes,this leads to a new proof of the Hausdorff dimension formula for SLE.  We also consider SLE$_\kappa(\rho)$ processes, which were originally onlydefined for $\rho > -2$, but which can also be defined for $\rho \leq -2$ usingL\'evy compensation. The range of an SLE$_\kappa(\rho)$ is qualitativelydifferent when $\rho \leq -2$. In particular, these curves areself-intersecting for $\kappa < 4$ and double points are dense, while ordinarySLE$_\kappa$ is simple. It was previously shown (Miller-Sheffield, 2016) thatcertain SLE$_\kappa(\rho)$ curves agree in law with certain light cones.Combining this with other known results, we obtain a general formula for theHausdorff dimension of SLE$_\kappa(\rho)$ for all values of $\rho$.  Finally, we show that the Hausdorff dimension of the so-called SLE$_\kappa$fan is the same as that of ordinary SLE$_\kappa$.

Dimension of the SLE light cone, the SLE fan, and SLE$_\kappa(\rho)$ for $\kappa \in (0,4)$ and $\rho \in [\tfrac{\kappa}{2}-4,-2)$