The Near-Critical Two-Point Function and the Torus Plateau for Weakly Self-avoiding Walk in High Dimensions

We use the lace expansion to study long-distance decay of two-point function weakly self-avoiding walk on integer lattice $$\mathbb {Z}^d$$ in dimensions $$d>4$$ , vicinity critical point, and prove an upper bound $$|x|^{-(d-2)}\exp [-c|x|/\xi ]$$ where correlation length $$\xi $$ has a square root divergence at point. As application, we that for discrete torus $$d{>}4$$ “plateau.” also discuss significance consequences plateau analysis behaviour torus.