Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 40, 2015, 601-616

CONVERGENCE OF AN ALGORITHM SIMULATING LOEWNER CURVES

Huy Tran

University of Washington, Department of Mathematics
Box 354350, Seattle, WA 98195-4350, U.S.A.; tvhuy 'at' math.washington.edu

Abstract. The development of Schramm-Loewner evolution (SLE) as the scaling limits of discrete models from statistical physics makes direct simulation of SLE an important task. The most common method, suggested by Marshall and Rohde [MR05], is to sample Brownian motion at discrete times, interpolate appropriately in between and solve explicitly the Loewner equation with this approximation. This algorithm always produces piecewise smooth non self-intersecting curves whereas SLE_κ has been proven to be simple for κ ∈ [0,4], self-touching for κ ∈ (4,8) and space-filling for κ ≥ 8. In this paper we show that this sequence of curves converges to SLE_κ for all κ ≠ 8 by giving a condition on deterministic driving functions to ensure the sup-norm convergence of simulated curves when we use this algorithm.

2010 Mathematics Subject Classification: Primary 30C30; Secondary 60J67.

Key words: Loewner differential equation, zipper algorithm.

Reference to this article: H. Tran: Convergence of an algorithm simulating Loewner curves. Ann. Acad. Sci. Fenn. Math. 40 (2015), 601-616.

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doi:10.5186/aasfm.2015.4037