Abstract
The determination of the Hausdorff dimension of the scaling limit of loop-erased random walk is closely related to the study of the one-point function of loop-erased random walk, i.e., the probability a loop-erased random walk passes through a given vertex. Recent work in the theoretical physics literature has investigated the Hausdorff dimension of loop-erased random walk in three dimensions by applying field theory techniques to study spin systems that heuristically encode the one-point function of loop-erased random walk. Inspired by this, we introduce two different spin systems whose correlation functions can be rigorously shown to encode the one-point function of loop-erased random walk.
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Acknowledgements
The authors thank K. J. Wiese for bringing this problem to our attention. We also thank the referees for their thorough and helpful reports on a previous version of this article. TH was at the University of Bristol when this work was carried out, and was supported by EPSRC Grant EP/P003656/1.
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Communicated by Yvan Velenik.
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Helmuth, T., Shapira, A. Loop-Erased Random Walk as a Spin System Observable. J Stat Phys 181, 1306–1322 (2020). https://doi.org/10.1007/s10955-020-02628-7
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DOI: https://doi.org/10.1007/s10955-020-02628-7