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

2023/10/26 by Yucheng Liu, Liu, Yucheng · 1 citation
Mathematics · Physics and Astronomy · #60K35 #82B27 #82B41 #FOS: Mathematics #FOS: Physical sciences #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics

paper · pdf · doi:10.48550/arxiv.2310.17321

openalex publication_date 2023/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove a sufficient condition for the two-point function of a statistical mechanical model on ℤ^d, d > 2, to be bounded uniformly near a critical point by |x|^-(d-2) exp [ -c|x| / ξ], where ξ 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 ℤ^d or on the torus.

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