Comments on: From the Littlewood-Offord problem to the Circular Law: universality of the spectral distribution of random matrices http://terrytao.wordpress.com/2008/10/18/from-the-littlewood-offord-problem-to-the-circular-law-universality-of-the-spectral-distribution-of-random-matrices/ Updates on my research and expository papers, discussion of open problems, and other maths-related topics. By Terence Tao Wed, 19 Jun 2013 16:27:44 +0000 hourly 1 http://wordpress.com/ By: Random matrices: The distribution of the smallest singular values « What’s new http://terrytao.wordpress.com/2008/10/18/from-the-littlewood-offord-problem-to-the-circular-law-universality-of-the-spectral-distribution-of-random-matrices/#comment-36305 Wed, 04 Mar 2009 16:53:13 +0000 http://terrytao.wordpress.com/?p=855#comment-36305 [...] with our earlier paper on the universality of the circular law, we do not attempt to prove (1) for general ensembles such as the Bernoulli ensemble (say) [...]

]]> By: Terence Tao http://terrytao.wordpress.com/2008/10/18/from-the-littlewood-offord-problem-to-the-circular-law-universality-of-the-spectral-distribution-of-random-matrices/#comment-33266 Sat, 25 Oct 2008 01:50:12 +0000 http://terrytao.wordpress.com/?p=855#comment-33266 Dear anonymous: thanks for the correction! We’ll include it in the next major revision of the article.

]]> By: anonymous http://terrytao.wordpress.com/2008/10/18/from-the-littlewood-offord-problem-to-the-circular-law-universality-of-the-spectral-distribution-of-random-matrices/#comment-33263 Fri, 24 Oct 2008 19:46:40 +0000 http://terrytao.wordpress.com/?p=855#comment-33263 dear prof. Tao

in the bottom of page 9 of the article
it is written: {(x_1,…,x_d) \in Z^d | M_i <= m_i <= M^’_i for all 1<=i<=d

you should replace the m_i with x_i

]]> By: Terence Tao http://terrytao.wordpress.com/2008/10/18/from-the-littlewood-offord-problem-to-the-circular-law-universality-of-the-spectral-distribution-of-random-matrices/#comment-33234 Wed, 22 Oct 2008 17:08:29 +0000 http://terrytao.wordpress.com/?p=855#comment-33234 Dear anonymous,

This is a very nice problem. The recent progress on understanding issues such as the effect of random noise on the invertibility of a matrix does support at a heuristic level the empirically verified observation that Gaussian elimination works very well in the presence of random noise, and may indeed help in giving a rigorous explanation of the latter in the future, but there are still significant technical issues to overcome before this is the case. The basic problem is that even if the original matrix A is described by an additive noise model, e.g. A=M+N where M is deterministic and N is a gaussian random matrix, after a few rounds of Gaussian elimination, the matrix that one gets from A is now given by a much more nonlinear random model, and it is not clear how to use the existing technology to continue to guarantee that this new matrix reacts will to future pivoting operations with high probability. One possibility would be to construct some sort of “invariant measure” for the Gaussian elimination algorithm, but it is not obvious to me how one would build such a measure and be able to ensure that it is not singular, unless some algebraic miracle intervenes.

]]> By: Anonymous http://terrytao.wordpress.com/2008/10/18/from-the-littlewood-offord-problem-to-the-circular-law-universality-of-the-spectral-distribution-of-random-matrices/#comment-33228 Wed, 22 Oct 2008 08:23:02 +0000 http://terrytao.wordpress.com/?p=855#comment-33228 Four weeks ago you blogged about the “The Princeton Companion to Mathematics” and gave a link to its home page. I enjoyed reading some of the sample chapters given there (I guess I’m gonna buy the book – thanks for bringing it to my attention).
In the sample chapter about “Numerical Analysis” (http://press.princeton.edu/chapters/gowers/gowers_IV_21.pdf), section 4, the author Lloyd N. Trefethen mentions the still missing theoretical analysis of Gaussian elimination with pivoting. Any chance you can tackle this problem using the techniques from your article?

]]> By: Anonymous http://terrytao.wordpress.com/2008/10/18/from-the-littlewood-offord-problem-to-the-circular-law-universality-of-the-spectral-distribution-of-random-matrices/#comment-33171 Mon, 20 Oct 2008 20:05:11 +0000 http://terrytao.wordpress.com/?p=855#comment-33171 A power of 2 is missing in the |Y_i| terms in the sentence immediately above equation (3). [Corrected - T.]

]]> By: Terence Tao http://terrytao.wordpress.com/2008/10/18/from-the-littlewood-offord-problem-to-the-circular-law-universality-of-the-spectral-distribution-of-random-matrices/#comment-33115 Sun, 19 Oct 2008 00:08:52 +0000 http://terrytao.wordpress.com/?p=855#comment-33115 Oops, there was a typo: Y_j is supposed to be the j^th column of A^{-1}, not the j^th column of A.

]]> By: Anonymous http://terrytao.wordpress.com/2008/10/18/from-the-littlewood-offord-problem-to-the-circular-law-universality-of-the-spectral-distribution-of-random-matrices/#comment-33110 Sat, 18 Oct 2008 21:09:41 +0000 http://terrytao.wordpress.com/?p=855#comment-33110 It is not evident that inner product of Y_j with X_j is one. The expectation of the inner product is one, that is for sure. Then it is not clear whether the equalities hold with expectations or not. But if the goal is to make the reader interested in the field, it is achieved perfectly :)

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