Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Dimension of the SLE light cone, the SLE fan, and SLE$_\kappa(\rho)$ for
  $\kappa \in (0,4)$ and $\rho \in [\tfrac{\kappa}{2}-4,-2)$   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 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 the
Hausdorff 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 range
of 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 only
defined for $\rho > -2$, but which can also be defined for $\rho \leq -2$ using
L\'evy compensation. The range of an SLE$_\kappa(\rho)$ is qualitatively
different when $\rho \leq -2$. In particular, these curves are
self-intersecting for $\kappa < 4$ and double points are dense, while ordinary
SLE$_\kappa$ is simple. It was previously shown (Miller-Sheffield, 2016) that
certain SLE$_\kappa(\rho)$ curves agree in law with certain light cones.
Combining this with other known results, we obtain a general formula for the
Hausdorff 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$_\kappa$.

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The categories of this paper are: nlin
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The categories of this paper are: cs
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The categories of this paper are: hep-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-th
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The categories of this paper are: q-bio
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The categories of this paper are: supr-con
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The categories of this paper are: bayes-an
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The categories of this paper are: funct-an
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The categories of this paper are: solv-int
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The categories of this paper are: quant-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: econ
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The categories of this paper are: comp-gas
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The categories of this paper are: acc-phys
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The categories of this paper are: alg-geom