Near-Critical and Finite-Size Scaling for High-Dimensional Lattice Trees and Animals

IF 1.3 3区 物理与天体物理 Q3 PHYSICS, MATHEMATICAL
Yucheng Liu, Gordon Slade
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引用次数: 0

Abstract

We consider spread-out models of lattice trees and lattice animals on \({\mathbb {Z}}^d\), for d above the upper critical dimension \(d_{\textrm{c}}=8\). We define a correlation length and prove that it diverges as \((p_c-p)^{-1/4}\) at the critical point \(p_c\). Using this, we prove that the near-critical two-point function is bounded above by \(C|x|^{-(d-2)}\exp [-c(p_c-p)^{1/4}|x|]\). We apply the near-critical bound to study lattice trees and lattice animals on a discrete d-dimensional torus (with \(d > d_{\textrm{c}}\)) of volume V. For \(p_c-p\) of order \(V^{-1/2}\), we prove that the torus susceptibility is of order \(V^{1/4}\), and that the torus two-point function behaves as \(|x|^{-(d-2)} + V^{-3/4}\) and thus has a plateau of size \(V^{-3/4}\). The proofs require significant extensions of previous results obtained using the lace expansion.

Abstract Image

高维网格树和动物的近临界和有限大小扩展
我们考虑在\({\mathbb {Z}}^d\)上的晶格树和晶格动物的展开模型,对于高于上临界维\(d_{\textrm{c}}=8\)的d。我们定义了一个相关长度,并证明它在临界点\(p_c\)处发散为\((p_c-p)^{-1/4}\)。利用这一点,我们证明了近临界两点函数的上界为\(C|x|^{-(d-2)}\exp [-c(p_c-p)^{1/4}|x|]\)。我们应用近临界界研究了体积为v的离散d维环面(含有\(d > d_{\textrm{c}}\))上的格树和格动物。对于阶为\(V^{-1/2}\)的\(p_c-p\),我们证明了环面磁化率为\(V^{1/4}\)阶,并且证明了环面两点函数表现为\(|x|^{-(d-2)} + V^{-3/4}\),因此具有一个大小为\(V^{-3/4}\)的平台。这些证明需要对先前使用蕾丝展开得到的结果进行显著的扩展。
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来源期刊
Journal of Statistical Physics
Journal of Statistical Physics 物理-物理:数学物理
CiteScore
3.10
自引率
12.50%
发文量
152
审稿时长
3-6 weeks
期刊介绍: The Journal of Statistical Physics publishes original and invited review papers in all areas of statistical physics as well as in related fields concerned with collective phenomena in physical systems.
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