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Communications in Mathematical Physics

, Volume 372, Issue 2, pp 543–572 | Cite as

Critical Two-Point Function for Long-Range Models with Power-Law Couplings: The Marginal Case for \({d\ge d_{\rm c}}\)

  • Lung-Chi Chen
  • Akira SakaiEmail author
Article

Abstract

Consider the long-range models on \({\mathbb{Z}^d}\) of random walk, self-avoiding walk, percolation and the Ising model, whose translation-invariant 1-step distribution/coupling coefficient decays as \({|x|^{-d-\alpha}}\) for some \({\alpha > 0}\) . In the previous work (Chen and Sakai in Ann Probab 43:639–681, 2015), we have shown in a unified fashion for all \({\alpha\ne2}\) that, assuming a bound on the “derivative” of the \({n}\) -step distribution (the compound-zeta distribution satisfies this assumed bound), the critical two-point function \({G_{p_{\rm c}}(x)}\) decays as \({|x|^{\alpha\wedge2-d}}\) above the upper-critical dimension \({d_{\rm c}\equiv(\alpha\wedge2)m}\) , where m = 2 for self-avoiding walk and the Ising model and m = 3 for percolation. In this paper, we show in a much simpler way, without assuming a bound on the derivative of the n-step distribution, that \({G_{p_{\rm c}}(x)}\) for the marginal case α = 2 decays as \({|x|^{2-d}/\log|x|}\) whenever d ≥ dc (with a large spread-out parameter L). This solves the conjecture in Chen and Sakai (2015), extended all the way down to d = dc, and confirms a part of predictions in physics (Brezin et al. in J Stat Phys 157:855–868, 2014). The proof is based on the lace expansion and new convolution bounds on power functions with log corrections.

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Notes

Acknowledgments

The work of AS was supported by JSPS KAKENHI Grant Number 18K03406. The work of LCC was supported by the Grant MOST 107-2115-M-004-004-MY2. We are grateful to the Institute of Mathematics and Mathematics Research Promotion Center (MRPC) of Academia Sinica, as well as the National Center for Theoretical Sciences (NCTS) at National Taiwan University, for providing us with comfortable working environment in multiple occasions. We would also like to thank the referees for their comments to improve presentation of this paper.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNational Chengchi UniversityTaipeiTaiwan
  2. 2.Mathematics DivisionNational Center for Theoretical SciencesTaipeiTaiwan
  3. 3.Faculty of ScienceHokkaido UniversitySapporo, HokkaidoJapan

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