二聚体模型与一般权重的随机格点置换
Dimer model and random lattice permutations with general weights
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- Johannes Gutenberg-Universität Mainz（约翰内斯·古腾堡美因茨大学）
- Sapienza Università di Roma（罗马智慧大学）
- NYU Shanghai（纽约大学上海分校）
- NYU-ECNU Math Institute（纽约大学华东师范大学数学研究所）
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中文总结 AI 辅助
本文在统一框架下研究二聚体模型与随机格点置换，证明有限或无限范围权重下长程有序与宏观环的出现，并给出不存在性准则。
中文摘要 AI 辅助
二聚体模型和随机格点置换是概率论、组合数学和数学物理交汇处的两个基本对象。我们在$\mathbb{Z}^d$中的有限周期盒内，于一个统一框架下研究这些模型。对于二聚体模型，连接任意顶点的边带有依赖于其相对位移的权重，二聚体构型通过其占据的边来加权。叠加两个独立的完美匹配得到双二聚体模型，其构型是不相交环的集合。相反，置换通过其跳跃的空间位移来加权。对于有限或无限范围的广泛权重类别，我们证明了长程有序和宏观环的出现。这将最近邻结果推广到任意范围的边和跳跃。特别地，长程权重在维度$d=1,2$中已经产生长程有序和宏观环。在二维中，这种行为与最近邻模型有质的区别。我们用尖锐的不存在性准则补充这些结果。
英文摘要
The dimer model and random lattice permutations are two fundamental objects at the interface of probability, combinatorics, and mathematical physics. We study these models on finite periodic boxes in $\mathbb Z^d$ within a common framework. For the dimer model, edges connecting arbitrary vertices carry a weight which depends on their relative displacement and dimer configurations are weighted through their occupied edges. Superimposing two independent perfect matchings gives the double-dimer model, whose configurations are collections of disjoint loops. Permutations, instead, are weighted through the spatial displacement of their jumps. For broad classes of weights of finite or infinite range we prove long-range order and the occurrence of macroscopic loops. This extends nearest-neighbour results to arbitrary-range edges and jumps. In particular, long-range weights yield long-range order and macroscopic loops already in dimensions $d=1,2$. In dimension two, this behaviour is qualitatively different from that of the nearest-neighbour model. We complement these results with sharp absence criteria.
