The scaling limit of random cubic planar graphs

IF 1 2区数学 Q1 MATHEMATICS

Journal of the London Mathematical Society-Second Series Pub Date : 2024-11-05 DOI:10.1112/jlms.70018

Benedikt Stufler

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引用次数: 0

Abstract

We study the random cubic planar graph $C_{n}$ with an even number $n$ of vertices. We show that the Brownian map arises as Gromov–Hausdorff–Prokhorov scaling limit of $C_{n}$ as $n \in 2 N$ tends to infinity, after rescaling distances by $γ n^{- 1 / 4}$ for a specific constant $γ > 0$ . This is the first time a model of random graphs that are not embedded into the plane is shown to converge to the Brownian map. Our approach features a new method that allows us to relate distances on random graphs to first-passage percolation distances on their 3-connected core.

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随机立方平面图的缩放极限

我们研究了偶数个顶点的随机立方平面图 C n $\mathsf {C}_n$。我们证明，当 n ∈ 2 N $n \in 2 \mathbb {N}$趋于无穷大时，布朗映射作为 C n $mathsf {C}_n$ 的格罗莫夫-豪斯多夫-普罗霍罗夫缩放极限而出现，在对特定常数 γ > 0 $\gamma >0$ 对距离进行 γ n - 1 / 4 $\gamma n^{-1/4}$ 重缩放之后。这是首次证明未嵌入平面的随机图模型收敛于布朗图。我们的方法采用了一种新方法，使我们能够将随机图上的距离与它们的 3 连接核心上的第一通道渗流距离联系起来。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of the London Mathematical Society-Second Series 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

186

审稿时长

6-12 weeks

期刊介绍： The Journal of the London Mathematical Society has been publishing leading research in a broad range of mathematical subject areas since 1926. The Journal welcomes papers on subjects of general interest that represent a significant advance in mathematical knowledge, as well as submissions that are deemed to stimulate new interest and research activity.