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

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Abstract

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.

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  1. Department of Mathematics, University of Arizona, Tucson, AZ, 85721, USA

    Tom Kennedy

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  1. Tom Kennedy
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Kennedy, T. Transforming Fixed-Length Self-avoiding Walks into Radial SLE8/3 . J Stat Phys 146, 281–293 (2012). https://doi.org/10.1007/s10955-011-0406-5

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