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

IF 0.9 3区数学 Q3 MATHEMATICS, APPLIED

Mathematical Physics, Analysis and Geometry Pub Date : 2023-02-17 DOI:10.1007/s11040-023-09447-8

Gordon Slade

引用次数: 7

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.

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高维弱自避行走的近临界两点函数和环面平台

利用蕾丝展开的方法，研究了临界点附近的整数格\(\mathbb {Z}^d\)(维度\(d>4\))上弱自回避行走的两点函数的远距离衰减，并证明了相关长度\(\xi \)在临界点处具有平方根散度的上界\(|x|^{-(d-2)}\exp [-c|x|/\xi ]\)。作为一个应用，我们证明了在一维\(d{>}4\)离散环面上弱自回避行走的两点函数具有“平台”。我们还讨论了平台对环面临界行为分析的意义和后果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Mathematical Physics, Analysis and Geometry 数学-物理：数学物理

CiteScore

2.10

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： MPAG is a peer-reviewed journal organized in sections. Each section is editorially independent and provides a high forum for research articles in the respective areas. The entire editorial board commits itself to combine the requirements of an accurate and fast refereeing process. The section on Probability and Statistical Physics focuses on probabilistic models and spatial stochastic processes arising in statistical physics. Examples include: interacting particle systems, non-equilibrium statistical mechanics, integrable probability, random graphs and percolation, critical phenomena and conformal theories. Applications of probability theory and statistical physics to other areas of mathematics, such as analysis (stochastic pde''s), random geometry, combinatorial aspects are also addressed. The section on Quantum Theory publishes research papers on developments in geometry, probability and analysis that are relevant to quantum theory. Topics that are covered in this section include: classical and algebraic quantum field theories, deformation and geometric quantisation, index theory, Lie algebras and Hopf algebras, non-commutative geometry, spectral theory for quantum systems, disordered quantum systems (Anderson localization, quantum diffusion), many-body quantum physics with applications to condensed matter theory, partial differential equations emerging from quantum theory, quantum lattice systems, topological phases of matter, equilibrium and non-equilibrium quantum statistical mechanics, multiscale analysis, rigorous renormalisation group.