The singularity probability of a random symmetric matrix is exponentially small

IF 3.5 1区数学 Q1 MATHEMATICS

Journal of the American Mathematical Society Pub Date : 2024-01-19 DOI:10.1090/jams/1042

Marcelo Campos, Matthew Jenssen, Marcus Michelen, Julian Sahasrabudhe

{"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":null,"pages":null},"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}

引用次数: 0

Abstract

Let A A be drawn uniformly at random from the set of all n × n n\times n symmetric matrices with entries in { − 1 , 1 } \{-1,1\} . We show that \[ P ( det ( A ) = 0 ) ⩽ e − c n , \mathbb {P}( \det (A) = 0 ) \leqslant e^{-cn}, \] where c > 0 c>0 is an absolute constant, thereby resolving a long-standing conjecture.

查看原文本刊更多论文

随机对称矩阵的奇异概率是指数级小的

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。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Journal of the American Mathematical Society 数学-数学

CiteScore

7.60

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： All articles submitted to this journal are peer-reviewed. The AMS has a single blind peer-review process in which the reviewers know who the authors of the manuscript are, but the authors do not have access to the information on who the peer reviewers are. This journal is devoted to research articles of the highest quality in all areas of pure and applied mathematics.