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Probability Theory and Related Fields

, Volume 171, Issue 3–4, pp 1093–1155 | Cite as

Green’s functions for chordal SLE curves

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Abstract

For a chordal SLE\(_\kappa \) (\(\kappa \in (0,8)\)) curve in a domain D, the n-point Green’s function valued at distinct points \(z_1,\dots ,z_n\in D\) is defined to be
$$\begin{aligned} G(z_1,\dots ,z_n)=\lim _{r_1,\dots ,r_n\downarrow 0} \prod _{k=1}^n r_k^{d-2} \mathbb {P}[{{\mathrm{dist}}}(\gamma ,z_k)<r_k,1\le k\le n], \end{aligned}$$
where \(d=1+\frac{\kappa }{8}\) is the Hausdorff dimension of SLE\(_\kappa \), provided that the limit converges. In this paper, we will show that such Green’s functions exist for any finite number of points. Along the way we provide the rate of convergence and modulus of continuity for Green’s functions as well. Finally, we give up-to-constant bounds for them.

Keywords

Chordal SLE Two-sided SLE Green’s function 

Mathematics Subject Classification

60G 30C 

Notes

Acknowledgements

The authors acknowledge Gregory Lawler, Brent Werness and Julien Dubédat for helpful discussions. Dapeng Zhan’s work is partially supported by a grant from NSF (DMS-1056840) and a grant from the Simons Foundation (#396973).

Supplementary material

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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Michigan State UniversityEast LansingUSA

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