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Monte Carlo Study of Four-Dimensional Self-avoiding Walks of up to One Billion Steps

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Abstract

We study self-avoiding walks on the four-dimensional hypercubic lattice via Monte Carlo simulations of walks with up to one billion steps. We study the expected logarithmic corrections to scaling, and find convincing evidence in support the scaling form predicted by the renormalization group, with an estimate for the power of the logarithmic factor of 0.2516(14), which is consistent with the predicted value of 1/4. We also characterize the behaviour of the pivot algorithm for sampling four dimensional self-avoiding walks, and conjecture that the probability of a pivot move being successful for an N-step walk is \(O([ \log N ]^{-1/4})\).

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Notes

  1. E.g., in our case we may choose which how many terms to include in our fits, whereas for differential approximant analyses of series it is possible to vary the order of the differential equation.

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Acknowledgements

Support from the Australian Research Council under the Future Fellowship scheme (Project Number FT130100972) and Discovery scheme (Project Number DP140101110) is gratefully acknowledged.

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Authors and Affiliations

  1. Department of Mathematics, Swinburne University of Technology, P.O. Box 218, Hawthorn, VIC, 3122, Australia

    Nathan Clisby

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  1. Nathan Clisby
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Correspondence to Nathan Clisby.

A Data

A Data

See Tables 1 and 2.

Table 1 Monte Carlo estimates of \(\langle R_{\mathrm {E}}^2 \rangle \), \(\langle R_{\mathrm {G}}^2 \rangle \), and \(\langle R_{\mathrm {E}}^2 \rangle / \langle R_{\mathrm {G}}^2 \rangle \)
Full size table
Table 2 Monte Carlo estimates of the probability of a pivot move being successful, f, and the mean CPU time per pivot attempt
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Clisby, N. Monte Carlo Study of Four-Dimensional Self-avoiding Walks of up to One Billion Steps. J Stat Phys 172, 477–492 (2018). https://doi.org/10.1007/s10955-018-2049-2

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