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 $|x{|}^{-(d+\hat{I}\pm )}$ where $0<\hat{I}\pm \le 2$, 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 $0<\hat{I}\pm <2$, we prove upper and lower bounds of the form $|x{|}^{-(d-\hat{I}\pm )}\min \{1,(p_c-p{)}^{-2}|x{|}^{-2\hat{I}\pm }\}$ for the two-point function near the critical point $p_c$. For $\hat{I}\pm =2$, 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.

18 pages. Minor edits. To appear in Electron. J. Probab