TY - JOUR

T1 - The length of an SLE-Monte Carlo studies

AU - Kennedy, Tom

N1 - Funding Information:
Acknowledgements Visits to the Kavli Institute for Theoretical Physics and the Banff International Research Station made possible many useful interactions. The author thanks David Brydges, Greg Lawler, Don Marshall, Daniel Meyer, Yuval Peres, Stephen Rohde, Oded Schramm, Wendelin Werner and Peter Young for useful discussions. In particular, the definition of fractal variation we use grew out of these discussions. This research was supported in part by the National Science Foundation under grants PHY99-07949 (KITP) and DMS-0501168 (TK).

PY - 2007/9

Y1 - 2007/9

N2 - 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 κ. 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.

AB - 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 κ. 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.

KW - Fractal variation

KW - Natural parametrization

KW - Schramm-Loewner evolution

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U2 - 10.1007/s10955-007-9375-0

DO - 10.1007/s10955-007-9375-0

M3 - Article

AN - SCOPUS:34548570078

VL - 128

SP - 1263

EP - 1277

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 6

ER -