论文标题
统一的无限立方平面图
The Uniform Infinite Cubic Planar Graph
论文作者
论文摘要
我们证明，随机简单的立方平面图$ \ mathsf {c} _n $带有偶数$ n $的Vertices，将新颖的统一无限立方平面图（UICPG）作为静置的局部限制。我们描述了如何通过应用于Angel和Schramm（Comm。Math。Phys。，2003）建立的一系列随机爆炸操作来构建极限。我们的主要技术引理是$ \ mathsf {c} _n $和一个模型之间的连续性关系，其中网络以$ \ mathsf {c} _n $的最大$ 3 $连接组件的链接插入。我们证明，最大$ 3 $连接的组件浓缩物的顶点数量为$κN$，$κ\约0.85085 $，订单$ n^{2/3} $的空气型波动。显示第二大组件的尺寸明显较小$ o_p（n^{2/3}）$。
We prove that the random simple cubic planar graph $\mathsf{C}_n$ with an even number $n$ of vertices admits a novel uniform infinite cubic planar graph (UICPG) as quenched local limit. We describe how the limit may be constructed by a series of random blow-up operations applied to the dual map of the type~III Uniform Infinite Planar Triangulation established by Angel and Schramm (Comm. Math. Phys., 2003). Our main technical lemma is a contiguity relation between $\mathsf{C}_n$ and a model where the networks inserted at the links of the largest $3$-connected component of $\mathsf{C}_n$ are replaced by independent copies of a specific Boltzmann network. We prove that the number of vertices of the largest $3$-connected component concentrates at $κn$ for $κ\approx 0.85085$, with Airy-type fluctuations of order $n^{2/3}$. The second-largest component is shown to have significantly smaller size $O_p(n^{2/3})$.
