{"title":"Ginibre矩阵中二次多项式的谱和伪谱","authors":"Nicholas A. Cook, A. Guionnet, Jonathan Husson","doi":"10.1214/21-aihp1225","DOIUrl":null,"url":null,"abstract":"For a fixed quadratic polynomial $\\mathfrak{p}$ in $n$ non-commuting variables, and $n$ independent $N\\times N$ complex Ginibre matrices $X_1^N,\\dots, X_n^N$, we establish the convergence of the empirical spectral distribution of $P^N =\\mathfrak{p}(X_1^N,\\dots, X_n^N)$ to the Brown measure of $\\mathfrak{p}$ evaluated at $n$ freely independent circular elements $c_1,\\dots, c_n$ in a non-commutative probability space. The main step of the proof is to obtain quantitative control on the pseudospectrum of $P^N$. Via the well-known linearization trick this hinges on anti-concentration properties for certain matrix-valued random walks, which we find can fail for structural reasons of a different nature from the arithmetic obstructions that were illuminated in works on the Littlewood--Offord problem for discrete scalar random walks.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":null,"pages":null},"PeriodicalIF":1.5000,"publicationDate":"2020-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"5","resultStr":"{\"title\":\"Spectrum and pseudospectrum for quadratic polynomials in Ginibre matrices\",\"authors\":\"Nicholas A. Cook, A. Guionnet, Jonathan Husson\",\"doi\":\"10.1214/21-aihp1225\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"For a fixed quadratic polynomial $\\\\mathfrak{p}$ in $n$ non-commuting variables, and $n$ independent $N\\\\times N$ complex Ginibre matrices $X_1^N,\\\\dots, X_n^N$, we establish the convergence of the empirical spectral distribution of $P^N =\\\\mathfrak{p}(X_1^N,\\\\dots, X_n^N)$ to the Brown measure of $\\\\mathfrak{p}$ evaluated at $n$ freely independent circular elements $c_1,\\\\dots, c_n$ in a non-commutative probability space. The main step of the proof is to obtain quantitative control on the pseudospectrum of $P^N$. Via the well-known linearization trick this hinges on anti-concentration properties for certain matrix-valued random walks, which we find can fail for structural reasons of a different nature from the arithmetic obstructions that were illuminated in works on the Littlewood--Offord problem for discrete scalar random walks.\",\"PeriodicalId\":42884,\"journal\":{\"name\":\"Annales de l Institut Henri Poincare D\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.5000,\"publicationDate\":\"2020-08-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"5\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Annales de l Institut Henri Poincare D\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1214/21-aihp1225\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"PHYSICS, MATHEMATICAL\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Annales de l Institut Henri Poincare D","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1214/21-aihp1225","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 5
摘要
对于$n$不可交换变量中的固定二次多项式$\mathfrak{p}$,以及$n$独立的$n \乘以n$复Ginibre矩阵$X_1^ n,\dots, X_n^ n$,我们建立了$ p ^ n =\mathfrak{p} $ (X_1^ n,\dots, X_n^ n)$的经验谱分布在$n$自由独立的圆元$c_1,\dots, c_n$处的Brown测度在非交换概率空间中的收敛性。证明的主要步骤是获得P^N$伪谱的定量控制。通过众所周知的线性化技巧,这取决于某些矩阵值随机漫步的反集中特性,我们发现,由于与离散标量随机漫步的Littlewood—offford问题中所揭示的算术障碍不同的结构原因,这种特性可能会失败。
Spectrum and pseudospectrum for quadratic polynomials in Ginibre matrices
For a fixed quadratic polynomial $\mathfrak{p}$ in $n$ non-commuting variables, and $n$ independent $N\times N$ complex Ginibre matrices $X_1^N,\dots, X_n^N$, we establish the convergence of the empirical spectral distribution of $P^N =\mathfrak{p}(X_1^N,\dots, X_n^N)$ to the Brown measure of $\mathfrak{p}$ evaluated at $n$ freely independent circular elements $c_1,\dots, c_n$ in a non-commutative probability space. The main step of the proof is to obtain quantitative control on the pseudospectrum of $P^N$. Via the well-known linearization trick this hinges on anti-concentration properties for certain matrix-valued random walks, which we find can fail for structural reasons of a different nature from the arithmetic obstructions that were illuminated in works on the Littlewood--Offord problem for discrete scalar random walks.