Dimension of the SLE Light Cone, the SLE Fan, and SLE κ(ρ) for κ∈ (0 , 4) and ρ∈ [κ2-4,-2)

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Miller, J 

Suppose that h is a Gaussian free field (GFF) on a planar domain. Fix κ∈(0,4). The SLEκ light cone L(θ) of h with opening angle θ∈[0,π] is the set of points reachable from a given boundary point by angle-varying flow lines of the (formal) vector field eih/χ, χ=2κκ2, with angles in [−θ2,θ2]. We derive the Hausdorff dimension of L(θ).

If θ=0 then L(θ) is an ordinary SLEκ curve (with κ<4); if θ=π then L(θ) is the range of an SLEκ curve (κ′=16/κ>4). In these extremes, this leads to a new proof of the Hausdorff dimension formula for SLE.

We also consider SLEκ(ρ) processes, which were originally only defined for ρ>−2, but which can also be defined for ρ≤−2 using Lévy compensation. The range of an SLEκ(ρ) is qualitatively different when ρ≤−2. In particular, these curves are self-intersecting for κ<4 and double points are dense, while ordinary SLEκ is simple. It was previously shown (Miller-Sheffield, 2016) that certain SLEκ(ρ) curves agree in law with certain light cones. Combining this with other known results, we obtain a general formula for the Hausdorff dimension of SLEκ(ρ) for all values of ρ.

Finally, we show that the Hausdorff dimension of the so-called SLEκ fan is the same as that of ordinary SLEκ.

4902 Mathematical Physics, 49 Mathematical Sciences
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Communications in Mathematical Physics
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Springer Science and Business Media LLC