TY - JOUR
T1 - Numerical computations for the schramm-loewner evolution
AU - Kennedy, Tom
N1 - Funding Information:
Acknowledgements I thank Don Marshall and Stephen Rohde for useful discussions. Talks and interactions during visits to the Banff International Research Station in March and May of 2005 and to the Kavli Institute for Theoretical Physics in September, 2006 contributed to the research included in these notes. The opportunity to present this material at the 2008 Enrage summer school at IHP is warmly acknowledged. This research was supported in part by the National Science Foundation under grants DMS-0201566 and DMS-0501168.
PY - 2009/12
Y1 - 2009/12
N2 - We review two numerical methods related to the Schramm-Loewner evolution (SLE). The first simulates SLE itself. More generally, it finds the curve in the half-plane that results from the Loewner equation for a given driving function. The second method can be thought of as the inverse problem. Given a simple curve in the half-plane it computes the driving function in the Loewner equation. This algorithm can be used to test if a given random family of curves in the half-plane is SLE by computing the driving process for the curves and testing if it is Brownian motion. More generally, this algorithm can be used to compute the driving process for random curves that may not be SLE. Most of the material presented here has appeared before. Our goal is to give a pedagogic review, illustrate some of the practical issues that arise in these computations and discuss some open problems.
AB - We review two numerical methods related to the Schramm-Loewner evolution (SLE). The first simulates SLE itself. More generally, it finds the curve in the half-plane that results from the Loewner equation for a given driving function. The second method can be thought of as the inverse problem. Given a simple curve in the half-plane it computes the driving function in the Loewner equation. This algorithm can be used to test if a given random family of curves in the half-plane is SLE by computing the driving process for the curves and testing if it is Brownian motion. More generally, this algorithm can be used to compute the driving process for random curves that may not be SLE. Most of the material presented here has appeared before. Our goal is to give a pedagogic review, illustrate some of the practical issues that arise in these computations and discuss some open problems.
KW - Random curves
KW - Schramm-Loewner evolution
KW - Simulation
KW - Zipper algorithm
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U2 - 10.1007/s10955-009-9866-2
DO - 10.1007/s10955-009-9866-2
M3 - Article
AN - SCOPUS:74649083449
SN - 0022-4715
VL - 137
SP - 839
EP - 856
JO - Journal of Statistical Physics
JF - Journal of Statistical Physics
IS - 5
ER -