二聚体模型中的一致正相关和$\mathbb{Z}^d$、$d > 2$上晶格排列中的相变
Uniformly positive correlations in the dimer model and phase transition in lattice permutations on $\mathbb{Z}^d$, $d > 2$, via reflection positivity
摘要：
我们的第一个主要结果是，在$\mathbb{Z}^d$的二聚体模型中的单体之间的相关性不衰减为零时，$d > 2$。这是关于二聚体模型中的相关性的第一个严格的结果，其维度大于大于2，并且表明该模型的行为与二维模型完全不同，在二维模型中，该模型是可积的，并且已知相关性以多项式方式衰减到零。这样的结果是隐含的，我们更一般的，第二个主要结果，其中国家的相变的发生在模型的晶格排列，这是有关的量子玻色气体。更确切地说，我们认为一个自我避免步行与晶格排列相互作用，我们证明，在制度的完全包装循环，这样的步行是“长”和它的端点之间的距离与直径呈线性增长的盒子。这些结果遵循从推导出的版本的红外线约束从一个新的一般概率设置，与彩色的循环和步行相互作用的网站和步行进入系统从一些“虚拟”顶点。
英文摘要：
Our first main result is that correlations between monomers in the dimer model in $\mathbb{Z}^d$ do not decay to zero when $d > 2$. This is the first rigorous result about correlations in the dimer model in dimensions greater than two and shows that the model behaves drastically differently than in two dimensions, in which case it is integrable and correlations are known to decay to zero polynomially. Such a result is implied by our more general, second main result, which states the occurrence of a phase transition in the model of lattice permutations, which is related to the quantum Bose gas. More precisely, we consider a self-avoiding walk interacting with lattice permutations and we prove that, in the regime of fully-packed loops, such a walk is `long' and the distance between its end-points grows linearly with the diameter of the box. These results follow from the derivation of a version of the infrared bound from a new general probabilistic settings, with coloured loops and walks interacting at sites and walks entering into the system from some `virtual' vertices.
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