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Finite-Order Correlation Length for Four-Dimensional Weakly Self-Avoiding Walk and \({|\varphi|^4}\) Spins

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Abstract

We study the four-dimensional n-component \({|\varphi|^4}\) spin model for all integers \({n \ge 1}\) and the four-dimensional continuous-time weakly self-avoiding walk which corresponds exactly to the case \({n=0}\) interpreted as a supersymmetric spin model. For these models, we analyse the correlation length of order p, and prove the existence of a logarithmic correction to mean-field scaling, with power \({\frac 12\frac{n+2}{n+8}}\), for all \({n \ge 0}\) and \({p > 0}\). The proof is based on an improvement of a rigorous renormalisation group method developed previously.

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Authors and Affiliations

  1. Department of Mathematics, Harvard University, 1 Oxford Street, Cambridge, MA, 02138, USA

    Roland Bauerschmidt

  2. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

    Gordon Slade, Alexandre Tomberg & Benjamin C. Wallace

Authors
  1. Roland Bauerschmidt
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  2. Gordon Slade
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  3. Alexandre Tomberg
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  4. Benjamin C. Wallace
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Corresponding author

Correspondence to Gordon Slade.

Additional information

Communicated by Abdelmalek Abdesselam.

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Bauerschmidt, R., Slade, G., Tomberg, A. et al. Finite-Order Correlation Length for Four-Dimensional Weakly Self-Avoiding Walk and \({|\varphi|^4}\) Spins. Ann. Henri Poincaré 18, 375–402 (2017). https://doi.org/10.1007/s00023-016-0499-0

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