均匀无限立方平面图

IF 1.5 2区 数学 Q2 STATISTICS & PROBABILITY
Bernoulli Pub Date : 2022-02-01 DOI:10.3150/22-bej1568
Benedikt Stufler
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引用次数: 5

摘要

我们证明了随机简单三次平面图$\mathsf{C}_n具有偶数个顶点的$n$允许一个新的一致无限三次平面图(UICPG)作为淬灭的局部极限。我们描述了如何通过应用于Angel和Schramm建立的~III型均匀无限平面三角剖分的对偶映射的一系列随机爆破操作来构造极限(Comm.Math.Phys.,2003)。我们的主要技术引理是$\mathsf之间的邻接关系{C}_n$和一个模型,其中网络插入$\mathsf最大的$3$连接组件的链接{C}_n$被特定Boltzmann网络的独立副本所取代。我们证明了最大$3$连通分量的顶点数集中在$\kappa n$,对于$\kapa\约0.85085$,具有次序为$n^{2/3}$的Airy型波动。第二大组件的大小$O_p(n^{2/3})$明显较小。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
The uniform infinite cubic planar graph
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 $\kappa n$ for $\kappa \approx 0.85085$, with Airy-type fluctuations of order $n^{2/3}$. The second-largest component is shown to have significantly smaller size $O_p(n^{2/3})$.
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来源期刊
Bernoulli
Bernoulli 数学-统计学与概率论
CiteScore
3.40
自引率
0.00%
发文量
116
审稿时长
6-12 weeks
期刊介绍: BERNOULLI is the journal of the Bernoulli Society for Mathematical Statistics and Probability, issued four times per year. The journal provides a comprehensive account of important developments in the fields of statistics and probability, offering an international forum for both theoretical and applied work. BERNOULLI will publish: Papers containing original and significant research contributions: with background, mathematical derivation and discussion of the results in suitable detail and, where appropriate, with discussion of interesting applications in relation to the methodology proposed. Papers of the following two types will also be considered for publication, provided they are judged to enhance the dissemination of research: Review papers which provide an integrated critical survey of some area of probability and statistics and discuss important recent developments. Scholarly written papers on some historical significant aspect of statistics and probability.
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