Gaussian free field light cones and SLE$_κ(ρ)$

Abstract

We derive a surprising correspondence between SLE$_κ(ρ)$ processes and light cones of the Gaussian free field (GFF). Recall that (one-sided, chordal, origin-seeded) SLE$_κ(ρ)$ processes are in some sense the simplest and most natural variants of the Schramm-Loewner evolution. They were originally defined only for $ρ> -2$, but one can use Lévy compensation to extend the definition to any $ρ> -2-\tfracκ{2}$ and to obtain qualitatively different curves. The triangle $T = \{(κ, ρ): (-2-\tfracκ{2})\vee (\tfracκ{2}-4) < ρ< -2 \}$ is the primary focus of this paper. When $(κ, ρ) \in T$, the SLE$_κ(ρ)$ curves are highly non-simple (and double points are dense) even though $κ< 4$. Let $h$ be an instance of the GFF. Fix $κ\in (0,4)$ and $χ= 2/\sqrtκ - \sqrtκ/2$. Recall that an imaginary geometry ray is a flow line of $e^{i(h/χ+θ)}$ that looks locally like SLE$_κ$. The light cone with parameter $θ\in [0, π]$ is the set of points reachable from the origin by a sequence of rays with angles in $[-θ/2, θ/2]$. When $θ=0$, the light cone looks like SLE$_κ$, and when $θ= π$ it looks like the range of an SLE$_{16/κ}$. We find that when $θ\in (0, π)$ the light cones are either fractal carpets with a dense set of holes or space-filling regions with no holes. We show that every non-space-filling light cone (with $θ\in (0,π]$ and $κ\in (0,4)$) agrees in law with the range of an SLE$_κ(ρ)$ process with $(κ, ρ) \in T$. Conversely, the range of any SLE$_κ(ρ)$ with $(κ,ρ) \in T$ agrees in law with a non-space-filling light cone. As a consequence, we obtain the first proof that these SLE$_κ(ρ)$ processes are continuous and show that they are natural path-valued functions of the GFF.

38 pages, 13 figures