TY - JOUR

T1 - Transforming Fixed-Length Self-avoiding Walks into Radial SLE8/3

AU - Kennedy, Tom

N1 - Funding Information:
Acknowledgements Oded Schramm’s suggestion to test if applying the conformal map to the fixed-length SAW would give radial SLE8/3 stimulated the author’s interest in this problem. The author has also benefited from discussions with Greg Lawler and Wendelin Werner. This research was supported in part by the National Science Foundation under grant DMS-0758649.

PY - 2012/1

Y1 - 2012/1

N2 - We conjecture a relationship between the scaling limit of the fixed-length ensemble of self-avoiding walks in the upper half plane and radial SLE8/3 in this half plane from 0 to i. The relationship is that if we take a curve from the fixed-length scaling limit of the SAW, weight it by a suitable power of the distance to the endpoint of the curve and apply the conformal map of the half plane that takes the endpoint to i, then we get the same probability measure on curves as radial SLE8/3. In addition to a non-rigorous derivation of this conjecture, we support it with Monte Carlo simulations of the SAW. Using the conjectured relationship between the SAW and radial SLE8/3, our simulations give estimates for both the interior and boundary scaling exponents. The values we obtain are within a few hundredths of a percent of the conjectured values.

AB - We conjecture a relationship between the scaling limit of the fixed-length ensemble of self-avoiding walks in the upper half plane and radial SLE8/3 in this half plane from 0 to i. The relationship is that if we take a curve from the fixed-length scaling limit of the SAW, weight it by a suitable power of the distance to the endpoint of the curve and apply the conformal map of the half plane that takes the endpoint to i, then we get the same probability measure on curves as radial SLE8/3. In addition to a non-rigorous derivation of this conjecture, we support it with Monte Carlo simulations of the SAW. Using the conjectured relationship between the SAW and radial SLE8/3, our simulations give estimates for both the interior and boundary scaling exponents. The values we obtain are within a few hundredths of a percent of the conjectured values.

KW - 2d self-avoiding walk

KW - Fixed length

KW - Radial SLE

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U2 - 10.1007/s10955-011-0406-5

DO - 10.1007/s10955-011-0406-5

M3 - Article

AN - SCOPUS:84655176741

VL - 146

SP - 281

EP - 293

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 2

ER -