High-dimensional long-range statistical mechanical models have random walk correlation functions

Abstract: We consider long-range percolation, Ising model, and self-avoiding walk on $\mathbb Z^d$, with couplings decaying like $|x|^{-(d+\alpha)}$ where $0 < \alpha \le 2$, above the upper critical dimensions. In the spread-out setting where the lace expansion applies, we show that the two-point function for each of these models exactly coincides with a random walk two-point function, up to a constant prefactor. Using this, for $0<\alpha < 2$, we prove upper and lower bounds of the form $|x|^{-(d-\alpha)} \min{ 1, (p_c - p)^{-2} |x|^{-2\alpha} }$ for the two-point function near the critical point $p_c$. For $\alpha=2$, we obtain a similar upper bound with logarithmic corrections. We also give a simple proof of the convergence of the lace expansion, assuming diagrammatic estimates.