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Critical Two-Point Function of the 4-Dimensional Weakly Self-Avoiding Walk

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Abstract

We prove \({|x|^{-2}}\) decay of the critical two-point function for the continuous-time weakly self-avoiding walk on \({\mathbb{Z}^{d}}\), in the upper critical dimension d = 4. This is a statement that the critical exponent \({\eta}\) exists and is equal to zero. Results of this nature have been proved previously for dimensions \({d \ge 5}\) using the lace expansion, but the lace expansion does not apply when d = 4. The proof is based on a rigorous renormalisation group analysis of an exact representation of the continuous-time weakly self-avoiding walk as a supersymmetric field theory. Much of the analysis applies more widely and has been carried out in a previous paper, where an asymptotic formula for the susceptibility is obtained. Here, we show how observables can be incorporated into the analysis to obtain a pointwise asymptotic formula for the critical two-point function. This involves perturbative calculations similar to those familiar in the physics literature, but with error terms controlled rigorously.

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Authors and Affiliations

  1. School of Mathematics, Institute for Advanced Study, Princeton, NJ, 08540, USA

    Roland Bauerschmidt

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

    David C. Brydges & Gordon Slade

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  1. Roland Bauerschmidt
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  2. David C. Brydges
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  3. Gordon Slade
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Correspondence to David C. Brydges.

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Communicated by M. Salmhofer

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Bauerschmidt, R., Brydges, D.C. & Slade, G. Critical Two-Point Function of the 4-Dimensional Weakly Self-Avoiding Walk. Commun. Math. Phys. 338, 169–193 (2015). https://doi.org/10.1007/s00220-015-2353-5

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