اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدThe circular law for random matrices
247
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a density for the distribution of entries. We assume that the entries have a finite moment of order larger than two and consider the case of sparse matrices. The results are based on previous work of Bai, Rudelson and the authors extending those results to a larger class of sparse matrices.
قيم البحث
اقرأ أيضاً
The eigenvalue distribution of the sum of two large Hermitian matrices, when one of them is conjugated by a Haar distributed unitary matrix, is asymptotically given by the free convolution of their spectral distributions. We prove that this convergen
ce also holds locally in the bulk of the spectrum, down to the optimal scales larger than the eigenvalue spacing. The corresponding eigenvectors are fully delocalized. Similar results hold for the sum of two real symmetric matrices, when one is conjugated by a Haar orthogonal matrix.
We extend the random characteristics approach to Wigner matrices whose entries are not required to have a normal distribution. As an application, we give a simple and fully dynamical proof of the weak local semicircle law in the bulk.
In the first part of this article, we proved a local version of the circular law up to the finest scale $N^{-1/2+ e}$ for non-Hermitian random matrices at any point $z in C$ with $||z| - 1| > c $ for any $c>0$ independent of the size of the matrix. U
nder the main assumption that the first three moments of the matrix elements match those of a standard Gaussian random variable after proper rescaling, we extend this result to include the edge case $ |z|-1=oo(1)$. Without the vanishing third moment assumption, we prove that the circular law is valid near the spectral edge $ |z|-1=oo(1)$ up to scale $N^{-1/4+ e}$.
We survey recent mathematical results about the spectrum of random band matrices. We start by exposing the Erd{H o}s-Schlein-Yau dynamic approach, its application to Wigner matrices, and extension to other mean-field models. We then introduce random
band matrices and the problem of their Anderson transition. We finally describe a method to obtain delocalization and universality in some sparse regimes, highlighting the role of quantum unique ergodicity.
This is a brief survey of classical and recent results about the typical behavior of eigenvalues of large random matrices, written for mathematicians and others who study and use matrices but may not be accustomed to thinking about randomness.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
