• DOI:10.1007/s00220-019-03385-9
• Corpus ID: 119120576

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

@article{Chen2018CriticalTF,
  title={Critical Two-Point Function for Long-Range Models with Power-Law Couplings: The Marginal Case for \$\$\{d\ge d\_\{\rm c\}\}\$\$d≥dc},
  author={Lung-Chi Chen and Akira Sakai},
  journal={Communications in Mathematical Physics},
  year={2018},
  pages={1-30}
}

• Lung-Chi Chen, A. Sakai
• Published 21 August 2018
• Mathematics, Physics
• Communications in Mathematical Physics

Consider the long-range models on $${\mathbb{Z}^d}$$Zd of random walk, self-avoiding walk, percolation and the Ising model, whose translation-invariant 1-step distribution/coupling coefficient decays as $${|x|^{-d-\alpha}}$$|x|-d-α for some $${\alpha > 0}$$α>0 . In the previous work (Chen and Sakai in Ann Probab 43:639–681, 2015), we have shown in a unified fashion for all $${\alpha\ne2}$$α≠2 that, assuming a bound on the “derivative” of the $${n}$$n -step distribution (the compound-zeta distribution satisfies this assumed bound), the critical two-point function $${G_{p_{\rm c}}(x)}$$Gpc(x) decays as $${|x|^{\alpha\wedge2-d}}$$|x|α∧2-d above the upper-critical dimension $${d_{\rm c}\equiv(\alpha\wedge2)m}$$dc≡(α∧2)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 $${G_{p_{\rm c}}(x)}$$Gpc(x) for the marginal case α = 2 decays as $${|x|^{2-d}/\log|x|}$$|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.