The Length of an SLE - Monte Carlo Studies

Details The Length of an SLE - Monte Carlo Studies The Length of an SLE - Monte Carlo Studies

2006-12-20

arxiv.org/abs/math/0612609v2

J. Statist. Phys. 128 (2007), 1263-1277

Mathematics

Probability

18 pages, 10 figures. Version 2 replaced the use of "nu" for the "growth exponent" by 1/d_H, where d_H is the Hausdorff dimens

Scientific paper

10.1007/s10955-007-9375-0

The scaling limits of a variety of critical two-dimensional lattice models are equal to the Schramm-Loewner evolution (SLE) for a suitable value of the parameter kappa. These lattice models have a natural parametrization of their random curves given by the length of the curve. This parametrization (with suitable scaling) should provide a natural parametrization for the curves in the scaling limit. We conjecture that this parametrization is also given by a type of fractal variation along the curve, and present Monte Carlo simulations to support this conjecture. Then we show by simulations that if this fractal variation is used to parametrize the SLE, then the parametrized curves have the same distribution as the curves in the scaling limit of the lattice models with their natural parametrization.

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