论文标题
部分可观测时空混沌系统的无模型预测
Macroscopic loops in the $3d$ double-dimer model
论文作者
论文摘要
双二聚体模型定义为两个图的两个独立分布的二聚体盖的叠加。它的配置可以看作是避开自我循环的不相交集合。我们的第一个结果是，在$ \ Mathbb {z}^d $，$ d> 2 $中，双二聚体模型中的循环是宏观的。这些表明这些行为的行为与两个维度的行为不同。特别是，我们表明，在一个大盒子的两个遥远点下，存在一个循环访问这两个点的均匀概率。我们的第二个结果涉及单体双二聚体模型，即在存在单体密度的情况下的双二聚体模型。这些是不允许任何循环触摸的顶点。该模型取决于一个控制单体密度的参数，即单体活性。从Betz和Taggi（2019）和Taggi（2021）中知道，单体活动的有限临界阈值存在，在此下面，在该临界阈值下面，在该临界活动中，在该临界阈值中，在该临界阈值中，在该临界活动中，在该临界阈值中，在此下，强迫通过系统的自我避免行走是宏观的。我们的论文表明，当$ d> 2 $时，如此关键的阈值严格是正面的。换句话说，即使存在单体正密度，自我避免的行走也是宏观的。
The double dimer model is defined as the superposition of two independent uniformly distributed dimer covers of a graph. Its configurations can be viewed as disjoint collections of self-avoiding loops. Our first result is that in $\mathbb{Z}^d$, $d>2$, the loops in the double dimer model are macroscopic. These are shown to behave qualitatively differently than in two dimensions. In particular, we show that, given two distant points of a large box, with uniformly positive probability there exists a loop visiting both points. Our second result involves the monomer double-dimer model, namely the double-dimer model in the presence of a density of monomers. These are vertices which are not allowed to be touched by any loop. This model depends on a parameter, the monomer activity, which controls the density of monomers. It is known from Betz and Taggi (2019) and Taggi (2021) that a finite critical threshold of the monomer activity exists, below which a self-avoiding walk forced through the system is macroscopic. Our paper shows that, when $d >2$, such a critical threshold is strictly positive. In other words, the self-avoiding walk is macroscopic even in the presence of a positive density of monomers.
