The near-critical two-point function for weakly self-avoiding walk in high dimensions Article Swipe
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We use the lace expansion to study the long-distance decay of the two-point function of weakly self-avoiding walk on the integer lattice $\mathbb{Z}^d$ in dimensions $d>4$, in the vicinity of the critical point, and prove an upper bound $|x|^{-(d-2)}\exp[-c|x|/\xi]$, where the correlation length $\xi$ has a square root divergence at the critical point. As one application, we prove that the two-point function for weakly self-avoiding walk on a discrete torus in dimensions $d>4$ has a ``plateau.'' A byproduct of the latter is an elementary proof of a similar plateau for simple random walk on a torus in dimensions $d>2$.
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