# 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

## References

- 1.Ahlfors, L.V.: Conformal Invariants: Topics in Geometric Function Theory. McGraw-Hill Book Co., New York (1973)zbMATHGoogle Scholar
- 2.Alberts, T., Kozdron, M.: Intersection probabilities for a chordal SLE path and a semicircle. Electron. Commun. Probab.
**13**, 448–460 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 3.Alberts, T., Kozdron, M., Lawler, G.: The Green function for the radial Schramm-Loewner evolution. J. Phys. A
**45**, 494015 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Beffara, V.: The dimension of SLE curves. Ann. Probab.
**36**, 1421–1452 (2008)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Beneš, C., Johansson Viklund, F., Kozdron, M.: On the rate of convergence of loop-erased random Walk to SLE\(_2\). Commun. Math. Phys
**318**(2), 307–354 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Friedrich, R., Werner, W.: Conformal restriction, highest-weight representations and SLE. Commun. Math. Phys.
**243**, 105–122 (2003)MathSciNetCrossRefzbMATHGoogle Scholar - 7.Jokela, N., Järvinen, M., Kytölä, K.: SLE boundary visits. Arxiv preprint http://arxiv.org/abs/1311.2297 (2015)
- 8.Lawler, G.: Conformally Invariant Processes in the Plane. Mathematical Surveys and Monographs, vol. 114. American Mathematical Society (2005)Google Scholar
- 9.Lawler, G.: Schramm-Loewner evolution. In: Sheffield, S., Spencer, T. (eds.) Statistical Mechanics. IAS/Park City Mathematical Series, pp. 231–295. AMS, Providence (2009)CrossRefGoogle Scholar
- 10.Lawler, G.: Minkowski content of the intersection of a Schramm-Loewner evolution (SLE) curve with the real line. J. Math. Soc. Jpn.
**67**, 1631–1669 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Lawler, G., Rezaei, M.: Minkowski content and natural parameterization for the Schramm-Loewner evolution. Ann. Probab.
**43**, 1082–1120 (2015)MathSciNetCrossRefzbMATHGoogle Scholar - 12.Lawler, G., Rezaei, M.: Up-to-constants bounds on the two-point Green’s function for SLE curves. Electron. Commun. Probab.
**20**(45), 1–13 (2015)Google Scholar - 13.Lawler, G., Schramm, O., Werner, W.: Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab.
**32**(1B), 939–995 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 14.Lawler, G., Werness, B.: Multi-point Green’s function for SLE and an estimate of Beffara. Ann. Probab.
**41**, 1513–1555 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 15.Lawler, G., Zhou, W.: SLE curves and natural parametrization. Ann. Probab.
**41**, 1556–1584 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 16.Rohde, S., Schramm, O.: Basic properties of SLE. Ann. Math.
**161**, 879–920 (2005)MathSciNetCrossRefzbMATHGoogle Scholar - 17.Rezaei, M., Zhan, D.: Higher moments of the natural parameterization for SLE curves. Ann. IHP.
**53**(1), 182–199 (2017)MathSciNetzbMATHGoogle Scholar - 18.Schramm, O.: Scaling limits of loop-erased random walks and uniform spanning trees, Israel. J. Math.
**118**, 221–288 (2000)MathSciNetzbMATHGoogle Scholar - 19.Zhan, D.: The scaling limits of planar LERW in finitely connected domains. Ann. Probab.
**36**, 467–529 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

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