关于斯皮尔伯格-邓猜想

IF 2.4 1区数学 Q1 MATHEMATICS

Geometric and Functional Analysis Pub Date : 2025-02-13 DOI:10.1007/s00039-025-00707-z

Ashwin Sah, Julian Sahasrabudhe, Mehtaab Sawhney

{"title":"关于斯皮尔伯格-邓猜想","authors":"Ashwin Sah, Julian Sahasrabudhe, Mehtaab Sawhney","doi":"10.1007/s00039-025-00707-z","DOIUrl":null,"url":null,"abstract":"Let M be an n×n matrix with iid subgaussian entries with mean 0 and variance 1 and let σn(M) denote the least singular value of M. We prove that $$ \\mathbb{P}\\big( \\sigma _{n}(M) \\leqslant \\varepsilon n^{-1/2} \\big) = (1+o(1)) \\varepsilon + e^{- \\Omega (n)} $$ for all 0⩽ε≪1. This resolves, up to a 1+o(1) factor, a seminal conjecture of Spielman and Teng.","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":"62 1","pages":""},"PeriodicalIF":2.4000,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the Spielman-Teng Conjecture\",\"authors\":\"Ashwin Sah, Julian Sahasrabudhe, Mehtaab Sawhney\",\"doi\":\"10.1007/s00039-025-00707-z\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let M be an n×n matrix with iid subgaussian entries with mean 0 and variance 1 and let σn(M) denote the least singular value of M. We prove that $$ \\\\mathbb{P}\\\\big( \\\\sigma _{n}(M) \\\\leqslant \\\\varepsilon n^{-1/2} \\\\big) = (1+o(1)) \\\\varepsilon + e^{- \\\\Omega (n)} $$ for all 0⩽ε≪1. This resolves, up to a 1+o(1) factor, a seminal conjecture of Spielman and Teng.\",\"PeriodicalId\":12478,\"journal\":{\"name\":\"Geometric and Functional Analysis\",\"volume\":\"62 1\",\"pages\":\"\"},\"PeriodicalIF\":2.4000,\"publicationDate\":\"2025-02-13\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Geometric and Functional Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00039-025-00707-z\",\"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":"Geometric and Functional Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00039-025-00707-z","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

设M是一个n×n矩阵，有iid个亚高斯分量，均值为0，方差为1，σn(M)表示M的最小奇异值。我们证明$$ \mathbb{P}\big( \sigma _{n}(M) \leqslant \varepsilon n^{-1/2} \big) = (1+o(1)) \varepsilon + e^{- \Omega (n)} $$对所有0≥ε≪1。这解决了Spielman和Teng的一个重要猜想，直到1+ 0(1)因子。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

查看原文本刊更多论文

On the Spielman-Teng Conjecture

Let M be an n×n matrix with iid subgaussian entries with mean 0 and variance 1 and let σ_n(M) denote the least singular value of M. We prove that

$$ \mathbb{P}\big( \sigma _{n}(M) \leqslant \varepsilon n^{-1/2} \big) = (1+o(1)) \varepsilon + e^{- \Omega (n)} $$

for all 0⩽ε≪1. This resolves, up to a 1+o(1) factor, a seminal conjecture of Spielman and Teng.

求助全文

通过发布文献求助，成功后即可免费获取论文全文。去求助

来源期刊

Geometric and Functional Analysis 数学-数学

CiteScore

3.70

自引率

4.50%

发文量

审稿时长

6-12 weeks

期刊介绍： Geometric And Functional Analysis (GAFA) publishes original research papers of the highest quality on a broad range of mathematical topics related to geometry and analysis. GAFA scored in Scopus as best journal in "Geometry and Topology" since 2014 and as best journal in "Analysis" since 2016. Publishes major results on topics in geometry and analysis. Features papers which make connections between relevant fields and their applications to other areas.