{"title":"随机 d 规则图谱，直至边缘","authors":"Jiaoyang Huang, Horng-Tzer Yau","doi":"10.1002/cpa.22176","DOIUrl":null,"url":null,"abstract":"<p>Consider the normalized adjacency matrices of random <i>d</i>-regular graphs on <i>N</i> vertices with fixed degree <math>\n <semantics>\n <mrow>\n <mi>d</mi>\n <mo>⩾</mo>\n <mn>3</mn>\n </mrow>\n <annotation>$d\\geqslant 3$</annotation>\n </semantics></math>. We prove that, with probability <math>\n <semantics>\n <mrow>\n <mn>1</mn>\n <mo>−</mo>\n <msup>\n <mi>N</mi>\n <mrow>\n <mo>−</mo>\n <mn>1</mn>\n <mo>+</mo>\n <mi>ε</mi>\n </mrow>\n </msup>\n </mrow>\n <annotation>$1-N^{-1+\\varepsilon }$</annotation>\n </semantics></math> for any <math>\n <semantics>\n <mrow>\n <mi>ε</mi>\n <mo>></mo>\n <mn>0</mn>\n </mrow>\n <annotation>$\\varepsilon >0$</annotation>\n </semantics></math>, the following two properties hold as <math>\n <semantics>\n <mrow>\n <mi>N</mi>\n <mo>→</mo>\n <mi>∞</mi>\n </mrow>\n <annotation>$N \\rightarrow \\infty$</annotation>\n </semantics></math> provided that <math>\n <semantics>\n <mrow>\n <mi>d</mi>\n <mo>⩾</mo>\n <mn>3</mn>\n </mrow>\n <annotation>$d\\geqslant 3$</annotation>\n </semantics></math>: (i) The eigenvalues are close to the classical eigenvalue locations given by the Kesten–McKay distribution. In particular, the extremal eigenvalues are concentrated with polynomial error bound in <i>N</i>, that is, <math>\n <semantics>\n <mrow>\n <msub>\n <mi>λ</mi>\n <mn>2</mn>\n </msub>\n <mo>,</mo>\n <mrow>\n <mo>|</mo>\n <msub>\n <mi>λ</mi>\n <mi>N</mi>\n </msub>\n <mo>|</mo>\n </mrow>\n <mo>⩽</mo>\n <mn>2</mn>\n <mo>+</mo>\n <msup>\n <mi>N</mi>\n <mrow>\n <mo>−</mo>\n <mi>c</mi>\n </mrow>\n </msup>\n </mrow>\n <annotation>$\\lambda _2, |\\lambda _N|\\leqslant 2+N^{-c}$</annotation>\n </semantics></math>. (ii) All eigenvectors of random <i>d</i>-regular graphs are completely delocalized.</p>","PeriodicalId":3,"journal":{"name":"ACS Applied Electronic Materials","volume":null,"pages":null},"PeriodicalIF":4.3000,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Spectrum of random d-regular graphs up to the edge\",\"authors\":\"Jiaoyang Huang, Horng-Tzer Yau\",\"doi\":\"10.1002/cpa.22176\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Consider the normalized adjacency matrices of random <i>d</i>-regular graphs on <i>N</i> vertices with fixed degree <math>\\n <semantics>\\n <mrow>\\n <mi>d</mi>\\n <mo>⩾</mo>\\n <mn>3</mn>\\n </mrow>\\n <annotation>$d\\\\geqslant 3$</annotation>\\n </semantics></math>. We prove that, with probability <math>\\n <semantics>\\n <mrow>\\n <mn>1</mn>\\n <mo>−</mo>\\n <msup>\\n <mi>N</mi>\\n <mrow>\\n <mo>−</mo>\\n <mn>1</mn>\\n <mo>+</mo>\\n <mi>ε</mi>\\n </mrow>\\n </msup>\\n </mrow>\\n <annotation>$1-N^{-1+\\\\varepsilon }$</annotation>\\n </semantics></math> for any <math>\\n <semantics>\\n <mrow>\\n <mi>ε</mi>\\n <mo>></mo>\\n <mn>0</mn>\\n </mrow>\\n <annotation>$\\\\varepsilon >0$</annotation>\\n </semantics></math>, the following two properties hold as <math>\\n <semantics>\\n <mrow>\\n <mi>N</mi>\\n <mo>→</mo>\\n <mi>∞</mi>\\n </mrow>\\n <annotation>$N \\\\rightarrow \\\\infty$</annotation>\\n </semantics></math> provided that <math>\\n <semantics>\\n <mrow>\\n <mi>d</mi>\\n <mo>⩾</mo>\\n <mn>3</mn>\\n </mrow>\\n <annotation>$d\\\\geqslant 3$</annotation>\\n </semantics></math>: (i) The eigenvalues are close to the classical eigenvalue locations given by the Kesten–McKay distribution. In particular, the extremal eigenvalues are concentrated with polynomial error bound in <i>N</i>, that is, <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>λ</mi>\\n <mn>2</mn>\\n </msub>\\n <mo>,</mo>\\n <mrow>\\n <mo>|</mo>\\n <msub>\\n <mi>λ</mi>\\n <mi>N</mi>\\n </msub>\\n <mo>|</mo>\\n </mrow>\\n <mo>⩽</mo>\\n <mn>2</mn>\\n <mo>+</mo>\\n <msup>\\n <mi>N</mi>\\n <mrow>\\n <mo>−</mo>\\n <mi>c</mi>\\n </mrow>\\n </msup>\\n </mrow>\\n <annotation>$\\\\lambda _2, |\\\\lambda _N|\\\\leqslant 2+N^{-c}$</annotation>\\n </semantics></math>. (ii) All eigenvectors of random <i>d</i>-regular graphs are completely delocalized.</p>\",\"PeriodicalId\":3,\"journal\":{\"name\":\"ACS Applied Electronic Materials\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.3000,\"publicationDate\":\"2023-09-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"ACS Applied Electronic Materials\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22176\",\"RegionNum\":3,\"RegionCategory\":\"材料科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"ENGINEERING, ELECTRICAL & ELECTRONIC\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Electronic Materials","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/cpa.22176","RegionNum":3,"RegionCategory":"材料科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"ENGINEERING, ELECTRICAL & ELECTRONIC","Score":null,"Total":0}
引用次数: 0
摘要
考虑 N 个顶点上具有固定度 d ⩾ 3 $d\geqslant 3$ 的随机 d-regular 图的归一化邻接矩阵。我们证明，对于任意 ε > 0 $\varepsilon >0$ ，只要 d ⩾ 3 $d\geqslant 3$，以下两个性质在 N → ∞ $N \rightarrow \infty$ 时成立：(i) 特征值接近凯斯顿-麦凯分布给出的经典特征值位置。特别是，极值特征值以 N 的多项式误差约束集中，即 λ 2 , | λ N | | ⩽ 2 + N - c $\lambda _2, |\lambda _N|leqslant 2+N^{-c}$ 。 (ii) 随机 d-regular 图形的所有特征向量都是完全非局部化的。
Spectrum of random d-regular graphs up to the edge
Consider the normalized adjacency matrices of random d-regular graphs on N vertices with fixed degree . We prove that, with probability for any , the following two properties hold as provided that : (i) The eigenvalues are close to the classical eigenvalue locations given by the Kesten–McKay distribution. In particular, the extremal eigenvalues are concentrated with polynomial error bound in N, that is, . (ii) All eigenvectors of random d-regular graphs are completely delocalized.
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