[Banach] Abstract of a paper by Mark Rudelson and Roman Vershynin

[Dale Alspach]

This is an announcement for the paper "The Littlewood-Offord Problem
and invertibility of random matrices" by Mark Rudelson and Roman
Vershynin.


Abstract: 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.

Archive classification: Probability; Functional Analysis

Mathematics Subject Classification: 15A52; 11P70

Remarks: 35 pages, no figures


Submitted from: vershynin at math.ucdavis.edu

The paper may be downloaded from the archive by web browser from
URL

 http://front.math.ucdavis.edu/math.PR/0703503

 or

 http://arXiv.org/abs/math.PR/0703503

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subject line

	 uget 0703503


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 to: math at arXiv.org.