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

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 $\alpha \ne 2$ 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 $|x{|}^{\alpha \wedge 2-d}$ above the upper-critical dimension $d_c\equiv (\alpha \wedge 2)m$, 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 $G_{p_c}(x)$ 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.

31 pages, 1 figure, 3 diagrams in equations

References in corpus (4)

• The crossover region between long-range and short-range interactions for the critical exponents
• Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk
• Critical two-point function for long-range $O(n)$ models below the upper critical dimension
• Asymptotic behavior of the gyration radius for long-range self-avoiding walk and long-range oriented percolation

Cited by in corpus (1)

• Sharp hierarchical upper bounds on the critical two-point function for long-range percolation on ${\mathbb{Z}}^d$