Two-curve Green's function for 2-SLE: the boundary case

2019/01/02 by Dapeng Zhan, Zhan, Dapeng · 1 citation
Mathematics · #30C #60D #FOS: Mathematics #Probability (math.PR) #math.PR #msc:30C #msc:60D

paper · pdf · doi:10.48550/arxiv.1901.00254

62 pages

arxiv created 2020/05/02 · arxiv updated 2020/05/05

Abstract

We prove that for κ∈(0,8), if (η₁,η₂) is a 2-SLE_κ pair in a simply connected domain D with an analytic boundary point z₀, then lim_{r→ 0⁺}r^-α ℙ[dist(z₀,η_j)<r,j=1,2] converges to a positive number for some α>0, which is called the two-curve Green's function. The exponent α equals \frac12κ-1 or 2(\frac12κ-1) depending on whether z₀ is one of the endpoints of η₁ and η₂. We also find the convergence rate and the exact formula of the Green's function up to a multiplicative constant. To derive these results, we construct two-dimensional diffusion processes and use orthogonal polynomials to obtain their transition density.

Cited by

• Time-reversal of multiple-force-point SLE_κ(\underlineρ) with all force points lying on the same side

Related

• Two-curve Green's function for 2-SLE: the interior case
• Green's function for chordal SLE curves
• Up-to-constants bounds on the two-point Green's function for SLE curves
• Green's function for cut points of chordal SLE attached with boundary arcs
• Existence of multi-point boundary Green's function for chordal Schramm-Loewner evolution (SLE)