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Consider the long-range models on $\mathbb{Z}^d$ of random walk,self-avoiding walk, percolation and the Ising model, whosetranslation-invariant 1-step distribution/coupling coefficient decays as$|x|^{-d-\alpha}$ for some $\alpha>0$. In the previous work (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 (thecompound-zeta distribution satisfies this assumed bound), the criticaltwo-point function $G_{p_c}(x)$ decays as $|x|^{\alpha\wedge2-d}$ above theupper-critical dimension $d_c\equiv(\alpha\wedge2)m$, where $m=2$ forself-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 thederivative of the $n$-step distribution, that $G_{p_c}(x)$ for the marginalcase $\alpha=2$ decays as $|x|^{2-d}/\log|x|$ whenever $d\ge d_c$ (with a largespread-out parameter $L$). This solves the conjecture in the previous work,extended all the way down to $d=d_c$, and confirms a part of predictions inphysics (Brezin, Parisi, Ricci-Tersenghi, J. Stat. Phys., 157, 855--868, 2014).The proof is based on the lace expansion and new convolution bounds on powerfunctions with log corrections.

Critical two-point function for long-range models with power-law couplings: The marginal case for $d\ge d_c$