Marcelo Campos, Matthew Jenssen, Marcus Michelen, Julian Sahasrabudhe
{"title":"随机对称矩阵的奇异概率是指数级小的","authors":"Marcelo Campos, Matthew Jenssen, Marcus Michelen, Julian Sahasrabudhe","doi":"10.1090/jams/1042","DOIUrl":null,"url":null,"abstract":"<p>Let <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"upper A\"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=\"application/x-tex\">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be drawn uniformly at random from the set of all <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"n times n\"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×<!-- × --></mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">n\\times n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> symmetric matrices with entries in <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"StartSet negative 1 comma 1 EndSet\"> <mml:semantics> <mml:mrow> <mml:mo fence=\"false\" stretchy=\"false\">{</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo fence=\"false\" stretchy=\"false\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\{-1,1\\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that <disp-formula content-type=\"math/mathml\"> \\[ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"double-struck upper P left-parenthesis det left-parenthesis upper A right-parenthesis equals 0 right-parenthesis less-than-or-slanted-equals e Superscript minus c n Baseline comma\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\"double-struck\">P</mml:mi> </mml:mrow> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mo movablelimits=\"true\" form=\"prefix\">det</mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>⩽<!-- ⩽ --></mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mi>c</mml:mi> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">\\mathbb {P}( \\det (A) = 0 ) \\leqslant e^{-cn},</mml:annotation> </mml:semantics> </mml:math> \\] </disp-formula> where <inline-formula content-type=\"math/mathml\"> <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" alttext=\"c greater-than 0\"> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\"application/x-tex\">c>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an absolute constant, thereby resolving a long-standing conjecture.</p>","PeriodicalId":54764,"journal":{"name":"Journal of the American Mathematical Society","volume":"30 1","pages":""},"PeriodicalIF":3.5000,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The singularity probability of a random symmetric matrix is exponentially small\",\"authors\":\"Marcelo Campos, Matthew Jenssen, Marcus Michelen, Julian Sahasrabudhe\",\"doi\":\"10.1090/jams/1042\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"upper A\\\"> <mml:semantics> <mml:mi>A</mml:mi> <mml:annotation encoding=\\\"application/x-tex\\\">A</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be drawn uniformly at random from the set of all <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"n times n\\\"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>×<!-- × --></mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">n\\\\times n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> symmetric matrices with entries in <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"StartSet negative 1 comma 1 EndSet\\\"> <mml:semantics> <mml:mrow> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">{</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo fence=\\\"false\\\" stretchy=\\\"false\\\">}</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\{-1,1\\\\}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We show that <disp-formula content-type=\\\"math/mathml\\\"> \\\\[ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"double-struck upper P left-parenthesis det left-parenthesis upper A right-parenthesis equals 0 right-parenthesis less-than-or-slanted-equals e Superscript minus c n Baseline comma\\\"> <mml:semantics> <mml:mrow> <mml:mrow> <mml:mi mathvariant=\\\"double-struck\\\">P</mml:mi> </mml:mrow> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mo movablelimits=\\\"true\\\" form=\\\"prefix\\\">det</mml:mo> <mml:mo stretchy=\\\"false\\\">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo stretchy=\\\"false\\\">)</mml:mo> <mml:mo>⩽<!-- ⩽ --></mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mi>c</mml:mi> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">\\\\mathbb {P}( \\\\det (A) = 0 ) \\\\leqslant e^{-cn},</mml:annotation> </mml:semantics> </mml:math> \\\\] </disp-formula> where <inline-formula content-type=\\\"math/mathml\\\"> <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\" alttext=\\\"c greater-than 0\\\"> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=\\\"application/x-tex\\\">c>0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an absolute constant, thereby resolving a long-standing conjecture.</p>\",\"PeriodicalId\":54764,\"journal\":{\"name\":\"Journal of the American Mathematical Society\",\"volume\":\"30 1\",\"pages\":\"\"},\"PeriodicalIF\":3.5000,\"publicationDate\":\"2024-01-19\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of the American Mathematical Society\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1090/jams/1042\",\"RegionNum\":1,\"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 American Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/jams/1042","RegionNum":1,"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
让 A A 从所有 n × n times n 对称矩阵的集合中均匀随机抽取，这些矩阵的条目在 { - 1 , 1 } 中。 \{-1,1\} .我们证明 \[ P ( det ( A ) = 0 )⩽ e - c n , \mathbb {P}( \det (A) = 0 )\leqslant e^{-cn}, \] 其中 c > 0 c>0 是一个绝对常量，从而解决了一个长期存在的猜想。
The singularity probability of a random symmetric matrix is exponentially small
Let AA be drawn uniformly at random from the set of all n×nn\times n symmetric matrices with entries in {−1,1}\{-1,1\}. We show that \[ P(det(A)=0)⩽e−cn,\mathbb {P}( \det (A) = 0 ) \leqslant e^{-cn}, \] where c>0c>0 is an absolute constant, thereby resolving a long-standing conjecture.
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