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A Non-intersecting Random Walk on the Manhattan Lattice and \({\hbox {SLE}}_{6}\)

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Abstract

We consider a random walk on the Manhattan lattice. The walker must follow the orientations of the bonds in this lattice, and the walker is not allowed to visit a site more than once. When both possible steps are allowed, the walker chooses between them with equal probability. The walks generated by this model are known to be related to interfaces for bond percolation on a square lattice. So it is natural to conjecture that the scaling limit is \(\hbox {SLE}_6\). We test this conjecture with Monte Carlo simulations of the random walk model and find strong support for the conjecture.

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Acknowledgements

An allocation of computer time from the UA Research Computing High Performance Computing (HPC) and High Throughput Computing (HTC) at the University of Arizona is gratefully acknowledged. Funding was provided by Directorate for Mathematical and Physical Sciences (Grant No. DMS-1500850).

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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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Kennedy, T. A Non-intersecting Random Walk on the Manhattan Lattice and \({\hbox {SLE}}_{6}\). J Stat Phys 174, 77–96 (2019). https://doi.org/10.1007/s10955-018-2176-9

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  • DOI: https://doi.org/10.1007/s10955-018-2176-9

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