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Mathematics > Probability
Title: The near-critical two-point function and the torus plateau for weakly self-avoiding walk in high dimensions
(Submitted on 31 Jul 2020 (v1), last revised 1 Sep 2022 (this version, v4))
Abstract: 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 an application, we prove that the two-point function for weakly self-avoiding walk on a discrete torus in dimensions $d>4$ has a "plateau." We also discuss the significance and consequences of the plateau for the analysis of critical behaviour on the torus.
Submission history
From: Gordon Slade [view email][v1] Fri, 31 Jul 2020 21:08:35 GMT (27kb)
[v2] Fri, 7 Aug 2020 19:23:32 GMT (27kb)
[v3] Mon, 26 Jul 2021 18:15:24 GMT (33kb)
[v4] Thu, 1 Sep 2022 16:37:18 GMT (33kb)
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