A general approach to massive upper bound for two-point function with application to self-avoiding walk torus plateau

We prove a sufficient condition for the two-point function of a statistical mechanical model on ${\mathbb{Z}}^d$, $d>2$, to be bounded uniformly near a critical point by $\mathrm{\mid }x{\mathrm{\mid }}^{-(d-2)}\exp [-c\mathrm{\mid }x\mathrm{\mid }\mathrm{/}\xi ]$, where $\xi$ is the correlation length. The condition is given in terms of a convolution equation satisfied by the two-point function, and we verify the condition for strictly self-avoiding walk in dimensions $d>4$ using the lace expansion. As an example application, we use the uniform bound to study the self-avoiding walk on a $d$-dimensional discrete torus with $d>4$, proving a ``plateau’’ of the torus two-point function, a result previously obtained for weakly self-avoiding walk in dimensions $d>4$ by Slade. Our method has the potential to be applied to other statistical mechanical models on ${\mathbb{Z}}^d$ or on the torus.