Abstract
In this paper, we study the convergence of Taylor approximations for the backward SLE maps near the origin. In addition, this result highlights the limitations of using stochastic Taylor methods for approximating \(SLE_{\kappa }\) traces. Due to the analytically tractable vector fields of the Loewner equation, we will show the Ninomiya–Victoir splitting is particularly well suited for SLE simulation. We believe that this is the first high order numerical method that has been successfully applied to \(SLE_{\kappa }\).
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Acknowledgements
We would like to thank Ilya Chevyrev, Dmitry Belyaev, Danyu Yang and Weijun Xi for useful suggestions and reading previous versions of this manuscript.
Funding
The first author was supported by the Department of Mathematical Sciences at the University of Bath and the DataSig programme under the EPSRC grant S026347/1. The second author was supported by the DataSig programme and Alan Turing Institute under the EPSRC Grant EP/N510129/1. The last author would like to acknowledge the support of ERC (Grant Agreement No.291244 Esig) between 2015–2017 at OMI Institute, EPSRC 1657722 between 2015-2018, Oxford Mathematical Department Grant and the EPSRC Grant EP/M002896/1 between 2018-2019. In addition, VM acknowledges the support of NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai.
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Foster, J., Lyons, T. & Margarint, V. An Asymptotic Radius of Convergence for the Loewner Equation and Simulation of \(SLE_{\kappa }\) Traces via Splitting. J Stat Phys 189, 18 (2022). https://doi.org/10.1007/s10955-022-02979-3
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DOI: https://doi.org/10.1007/s10955-022-02979-3