# 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 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.

*D*, the*n*-point Green’s function valued at distinct points \(z_1,\dots ,z_n\in D\) is defined to be$$\begin{aligned} G(z_1,\dots ,z_n)=\lim _{r_1,\dots ,r_n\downarrow 0} \prod _{k=1}^n r_k^{d-2} \mathbb {P}[{{\mathrm{dist}}}(\gamma ,z_k)<r_k,1\le k\le n], \end{aligned}$$

## 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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