{"title":"随机立方平面图的缩放极限","authors":"Benedikt Stufler","doi":"10.1112/jlms.70018","DOIUrl":null,"url":null,"abstract":"<p>We study the random cubic planar graph <span></span><math>\n <semantics>\n <msub>\n <mi>C</mi>\n <mi>n</mi>\n </msub>\n <annotation>$\\mathsf {C}_n$</annotation>\n </semantics></math> with an even number <span></span><math>\n <semantics>\n <mi>n</mi>\n <annotation>$n$</annotation>\n </semantics></math> of vertices. We show that the Brownian map arises as Gromov–Hausdorff–Prokhorov scaling limit of <span></span><math>\n <semantics>\n <msub>\n <mi>C</mi>\n <mi>n</mi>\n </msub>\n <annotation>$\\mathsf {C}_n$</annotation>\n </semantics></math> as <span></span><math>\n <semantics>\n <mrow>\n <mi>n</mi>\n <mo>∈</mo>\n <mn>2</mn>\n <mi>N</mi>\n </mrow>\n <annotation>$n \\in 2 \\mathbb {N}$</annotation>\n </semantics></math> tends to infinity, after rescaling distances by <span></span><math>\n <semantics>\n <mrow>\n <mi>γ</mi>\n <msup>\n <mi>n</mi>\n <mrow>\n <mo>−</mo>\n <mn>1</mn>\n <mo>/</mo>\n <mn>4</mn>\n </mrow>\n </msup>\n </mrow>\n <annotation>$\\gamma n^{-1/4}$</annotation>\n </semantics></math> for a specific constant <span></span><math>\n <semantics>\n <mrow>\n <mi>γ</mi>\n <mo>></mo>\n <mn>0</mn>\n </mrow>\n <annotation>$\\gamma &gt;0$</annotation>\n </semantics></math>. 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.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70018","citationCount":"0","resultStr":"{\"title\":\"The scaling limit of random cubic planar graphs\",\"authors\":\"Benedikt Stufler\",\"doi\":\"10.1112/jlms.70018\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>We study the random cubic planar graph <span></span><math>\\n <semantics>\\n <msub>\\n <mi>C</mi>\\n <mi>n</mi>\\n </msub>\\n <annotation>$\\\\mathsf {C}_n$</annotation>\\n </semantics></math> with an even number <span></span><math>\\n <semantics>\\n <mi>n</mi>\\n <annotation>$n$</annotation>\\n </semantics></math> of vertices. We show that the Brownian map arises as Gromov–Hausdorff–Prokhorov scaling limit of <span></span><math>\\n <semantics>\\n <msub>\\n <mi>C</mi>\\n <mi>n</mi>\\n </msub>\\n <annotation>$\\\\mathsf {C}_n$</annotation>\\n </semantics></math> as <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>n</mi>\\n <mo>∈</mo>\\n <mn>2</mn>\\n <mi>N</mi>\\n </mrow>\\n <annotation>$n \\\\in 2 \\\\mathbb {N}$</annotation>\\n </semantics></math> tends to infinity, after rescaling distances by <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>γ</mi>\\n <msup>\\n <mi>n</mi>\\n <mrow>\\n <mo>−</mo>\\n <mn>1</mn>\\n <mo>/</mo>\\n <mn>4</mn>\\n </mrow>\\n </msup>\\n </mrow>\\n <annotation>$\\\\gamma n^{-1/4}$</annotation>\\n </semantics></math> for a specific constant <span></span><math>\\n <semantics>\\n <mrow>\\n <mi>γ</mi>\\n <mo>></mo>\\n <mn>0</mn>\\n </mrow>\\n <annotation>$\\\\gamma &gt;0$</annotation>\\n </semantics></math>. 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.</p>\",\"PeriodicalId\":49989,\"journal\":{\"name\":\"Journal of the London Mathematical Society-Second Series\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.0000,\"publicationDate\":\"2024-11-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://onlinelibrary.wiley.com/doi/epdf/10.1112/jlms.70018\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the London Mathematical Society-Second Series\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1112/jlms.70018\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of the London Mathematical Society-Second Series","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1112/jlms.70018","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
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。
We study the random cubic planar graph with an even number of vertices. We show that the Brownian map arises as Gromov–Hausdorff–Prokhorov scaling limit of as tends to infinity, after rescaling distances by for a specific constant . 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.
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.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
