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Some Partial Results on the Convergence of Loop-Erased Random Walk to SLE(2) in the Natural Parametrization

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Abstract

We outline a strategy for showing convergence of loop-erased random walk on the \(\mathbb{Z}^{2}\) square lattice to SLE(2), in the supremum norm topology that takes the time parametrization of the curves into account. The discrete curves are parametrized so that the walker moves at a constant speed determined by the lattice spacing, and the SLE(2) curve has the recently introduced natural time parametrization. Our strategy can be seen as an extension of the one used by Lawler, Schramm, and Werner to prove convergence modulo time parametrization. The crucial extra step is showing that the expected occupation measure of the discrete curve, properly renormalized by the chosen time parametrization, converges to the occupation density of the SLE(2) curve, the so-called SLE Green’s function. Although we do not prove this convergence, we rigorously establish some partial results in this direction including a new loop-erased random walk estimate.

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Notes

  1. Lawler, G.F.: Private communication, 2013.

  2. Beneš, C., Lawler, G.F., Johansson Viklund, F.: In progress, 2013.

  3. Beneš, C., Lawler, G.F., Johansson Viklund, F.: In progress, 2013.

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Acknowledgements

The authors would like to express their gratitude to the Banff International Research Station for Mathematical Innovation and Discovery (BIRS), the Mathematical Sciences Research Institute (MSRI), and the Simons Center for Geometry and Physics where much of this work was carried out. In particular, the authors benefitted from participating in a Research in Teams at BIRS, as well as the Program on Random Spatial Processes at MSRI, and the Program on Conformal Geometry at the Simons Center. Thanks are owed to Ed Perkins, Martin Barlow, and Greg Lawler for useful discussions, and to three anonymous referees for several valuable comments. The research of the first two authors was supported in part by the Natural Sciences and Engineering Research Council (NSERC) of Canada.

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Authors and Affiliations

  1. Department of Mathematics, California Institute of Technology, Pasadena, CA, 91125, USA

    Tom Alberts

  2. Department of Mathematics & Statistics, University of Regina, Regina, SK, S4S 0A2, Canada

    Michael J. Kozdron

  3. Tradelink LLC, 71 South Wacker Drive, Suite 1900, Chicago, IL, 60606, USA

    Robert Masson

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  1. Tom Alberts
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  2. Michael J. Kozdron
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  3. Robert Masson
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Correspondence to Michael J. Kozdron.

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Alberts, T., Kozdron, M.J. & Masson, R. Some Partial Results on the Convergence of Loop-Erased Random Walk to SLE(2) in the Natural Parametrization. J Stat Phys 153, 119–141 (2013). https://doi.org/10.1007/s10955-013-0816-7

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