LAUSR

A 2- $$\hbox {SLE}_\kappa $$ SLE κ ( $$\kappa \in (0,8)$$ κ ∈ ( 0 , 8 ) ) is a pair of random curves $$(\eta _1,\eta _2)$$ ( η 1 , η 2 ) in a simply connected domain D connecting two pairs of boundary points such that conditioning on any curve, the other is a chordal $$\hbox {SLE}_\kappa $$ SLE κ curve in a complement domain. In this paper we prove that for any $$z_0\in D$$ z 0 ∈ D , the limit $$\lim _{r\rightarrow 0^+}r^{-\alpha _0} {\mathbb {P}}[{{\,\mathrm{dist}\,}}(z_0,\eta _j)