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Conformal Invariance of the Loop-Erased Percolation Explorer

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Abstract

We consider critical percolation on the triangular lattice in a bounded simply connected domain with boundary conditions that force an interface between two prescribed boundary points. We say the interface forms a “near-loop” when it comes within one lattice spacing of itself. We define a new curve by erasing these near-loops as we traverse the interface. Our Monte Carlo simulations of this model lead us to conclude that the scaling limit of this loop-erased percolation interface is conformally invariant and has fractal dimension 4 / 3. However, it is not \(\hbox {SLE}_{8/3}\). We also consider the process in which a near-loop is when the explorer comes within two lattice spacings of itself.

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Acknowledgements

This research was supported in part by NSF Grant DMS-1500850. An allocation of computer time from UA Research Computing at the University of Arizona is gratefully acknowledged.

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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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Correspondence to Tom Kennedy.

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Communicated by Eric A. Carlen.

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Kennedy, T. Conformal Invariance of the Loop-Erased Percolation Explorer. J Stat Phys 177, 1–19 (2019). https://doi.org/10.1007/s10955-019-02354-9

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