# 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} }

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…

## 3 Citations

### Correct Bounds on the Ising Lace-Expansion Coefficients

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The lace expansion for the Ising two-point function was successfully derived in (Sakai in Commun Math Phys 272:283–344, 2007, Proposition 1.1). It is an identity that involves an alternating series…

### Sharp hierarchical upper bounds on the critical two-point function for long-range percolation on Zd

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Consider long-range Bernoulli percolation on [Formula: see text] in which we connect each pair of distinct points x and y by an edge with probability 1 − exp(− β‖ x − y‖− d− α), where α > 0 is fixed…

### Power-law bounds for critical long-range percolation below the upper-critical dimension

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- 2021

We study long-range Bernoulli percolation on Zd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}…

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