Our first main result is that correlations between monomers in the dimer model in do not decay to 0 when . This is the first rigorous result about correlations in the dimer model in dimensions greater than 2 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 endpoints 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. © 2022 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

中文翻译：

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ℤd, d ≥ 3 中二聚体模型与宏观交互自回避行走的一致正相关

我们的第一个主要结果是二聚体模型中单体之间的相关性不会衰减到 0. 这是关于大于 2 维的二聚体模型中相关性的第一个严格结果，并表明该模型的行为与二维中的行为截然不同，在这种情况下，它是可积的，并且已知相关性会多项式衰减为零。我们更一般的第二个主要结果暗示了这样的结果，该结果表明在晶格排列模型中发生了相变，这与量子玻色气体有关。更准确地说，我们考虑了与晶格排列相互作用的自我避免游走，我们证明，在完全填充循环的情况下，这样的游走是“长的”，并且其端点之间的距离随着盒子的直径线性增长。这些结果来自于从新的一般概率设置推导红外界的一个版本，彩色循环和步行在站点交互，步行从一些“虚拟”顶点进入系统。© 2022 作者。Wiley Periodicals LLC 出版的关于纯数学和应用数学的通讯。