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

Abstract

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 $|x{|}^{-(d-2)}\exp [-c|x|/\xi ]$, 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.

23 pages, 1 figure. Editorial improvements throughout

References in corpus (6)

• Decay of correlations in nearest-neighbor self-avoiding walk, percolation, lattice trees and animals
• Critical two-point function of the 4-dimensional weakly self-avoiding walk
• Random-length Random Walks and Finite-size scaling in high dimensions
• Weakly self-avoiding walk on a high-dimensional torus
• Gaussian deconvolution and the lace expansion
• Near-critical and finite-size scaling for high-dimensional lattice trees and animals

Cited by in corpus (1)

• High-dimensional long-range statistical mechanical models have random walk correlation functions