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

TL;DR

This paper proves the decay rate of the critical two-point function for long-range models with power-law couplings at the marginal case, confirming physics predictions and extending previous results to the critical dimension.

Contribution

It simplifies the proof of decay behavior at the marginal case $oldsymbol{ ext{α=2}}$, removing the need for derivative bounds and confirming a physics conjecture.

Findings

Decay of $G_{p_c}(x)$ as $|x|^{2-d}/ ext{log}|x|$ for $d \\ge d_c$

Extension of decay results to the marginal case $\\alpha=2$

Validation of physics predictions for long-range models at criticality

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 (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 $G_{p_c}(x)$ decays as $|x|^{\alpha\wedge2-d}$ above the upper-critical dimension $d_c\equiv(\alpha\wedge2)m$, 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 $G_{p_c}(x)$ for the marginal case $\alpha=2$ decays as $|x|^{2-d}/\log|x|$ whenever $d\ge d_c$ (with a large spread-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 in physics (Brezin, Parisi, Ricci-Tersenghi, 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.