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Van Vu and I have just uploaded to the arXiv our survey paper “From the Littlewood-Offord problem to the Circular Law: universality of the spectral distribution of random matrices“, submitted to Bull. Amer. Math. Soc..  This survey recaps (avoiding most of the technical details) the recent work of ourselves and others that exploits the inverse theory for the Littlewood-Offord problem (which, roughly speaking, amounts to figuring out what types of random walks exhibit concentration at any given point), and how this leads to bounds on condition numbers, least singular values, and resolvents of random matrices; and then how the latter then leads to universality of the empirical spectral distributions (ESDs) of random matrices, and in particular to the circular law for the ESDs for iid random matrices with zero mean and unit variance (see my previous blog post on this topic, or my Lewis lectures).  We conclude by mentioning a few open problems in the subject.

While this subject does unfortunately contain a large amount of technical theory and detail, every so often we find a very elementary observation that simplifies the work required significantly.  One such observation is an identity which we call the negative second moment identity, which I would like to discuss here.    Let A be an $n \times n$ matrix; for simplicity we assume that the entries are real-valued.  Denote the n rows of A by $X_1,\ldots,X_n$, which we view as vectors in ${\Bbb R}^n$.  Let $\sigma_1(A) \geq \ldots \geq \sigma_n(A) \geq 0$ be the singular values of A. In our applications, the vectors $X_j$ are easily described (e.g. they might be randomly distributed on the discrete cube $\{-1,1\}^n$), but the distribution of the singular values $\sigma_j(A)$ is much more mysterious, and understanding this distribution is a key objective in this entire theory.
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