Critical two-point functions for long-range statistical-mechanical models in high dimensions

We consider long-range self-avoiding walk, percolation and the Ising model on $\mathbb{Z}^d$ that are defined by power-law decaying pair potentials of the form $D(x)\asymp|x|^{-d-α}$ with $α>0$. The upper-critical dimension $d_{\mathrm{c}}$ is $2(α\wedge2)$ for self-avoiding walk and the Ising model, and $3(α\wedge2)$ for percolation. Let $α\ne2$ and assume certain heat-kernel bounds on the $n$-step distribution of the underlying random walk. We prove that, for $d>d_{\mathrm{c}}$ (and the spread-out parameter sufficiently large), the critical two-point function $G_{p_{\mathrm{c}}}(x)$ for each model is asymptotically $C|x|^{α\wedge2-d}$, where the constant $C\in(0,\infty)$ is expressed in terms of the model-dependent lace-expansion coefficients and exhibits crossover between $α<2$ and $α>2$. We also provide a class of random walks that satisfy those heat-kernel bounds.

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