5,026 research outputs found
Invertibility of symmetric random matrices
We study n by n symmetric random matrices H, possibly discrete, with iid
above-diagonal entries. We show that H is singular with probability at most
exp(-n^c), and the spectral norm of the inverse of H is O(sqrt{n}).
Furthermore, the spectrum of H is delocalized on the optimal scale o(n^{-1/2}).
These results improve upon a polynomial singularity bound due to Costello, Tao
and Vu, and they generalize, up to constant factors, results of Tao and Vu, and
Erdos, Schlein and Yau.Comment: 53 pages. Minor corrections, changes in presentation. To appear in
Random Structures and Algorithm
Smoothed analysis of symmetric random matrices with continuous distributions
We study invertibility of matrices of the form where is an
arbitrary symmetric deterministic matrix, and is a symmetric random matrix
whose independent entries have continuous distributions with bounded densities.
We show that with high probability. The bound is
completely independent of . No moment assumptions are placed on ; in
particular the entries of can be arbitrarily heavy-tailed.Comment: Several very small revisions mad
Invertibility of random matrices: unitary and orthogonal perturbations
We show that a perturbation of any fixed square matrix D by a random unitary
matrix is well invertible with high probability. A similar result holds for
perturbations by random orthogonal matrices; the only notable exception is when
D is close to orthogonal. As an application, these results completely eliminate
a hard-to-check condition from the Single Ring Theorem by Guionnet, Krishnapur
and Zeitouni.Comment: 46 pages. A more general result on orthogonal perturbations of
complex matrices added. It rectified an inaccuracy in application to Single
Ring Theorem for orthogonal matrice
No-gaps delocalization for general random matrices
We prove that with high probability, every eigenvector of a random matrix is
delocalized in the sense that any subset of its coordinates carries a
non-negligible portion of its norm. Our results pertain to a wide
class of random matrices, including matrices with independent entries,
symmetric and skew-symmetric matrices, as well as some other naturally arising
ensembles. The matrices can be real and complex; in the latter case we assume
that the real and imaginary parts of the entries are independent.Comment: 45 page
The Littlewood-Offord Problem and invertibility of random matrices
We prove two basic conjectures on the distribution of the smallest singular
value of random n times n matrices with independent entries. Under minimal
moment assumptions, we show that the smallest singular value is of order
n^{-1/2}, which is optimal for Gaussian matrices. Moreover, we give a optimal
estimate on the tail probability. This comes as a consequence of a new and
essentially sharp estimate in the Littlewood-Offord problem: for i.i.d. random
variables X_k and real numbers a_k, determine the probability P that the sum of
a_k X_k lies near some number v. For arbitrary coefficients a_k of the same
order of magnitude, we show that they essentially lie in an arithmetic
progression of length 1/p.Comment: Introduction restructured, some typos and minor errors correcte
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