Van Vu and I have just uploaded to the arXiv our paper “Random matrices have simple spectrum“. Recall that an Image may be NSFW.
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Hermitian matrix is said to have simple eigenvalues if all of its Image may be NSFW.
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eigenvalues are distinct. This is a very typical property of matrices to have: for instance, as discussed in this previous post, in the space of all Image may be NSFW.
Clik here to view. Hermitian matrices, the space of matrices without all eigenvalues simple has codimension three, and for real symmetric cases this space has codimension two. In particular, given any random matrix ensemble of Hermitian or real symmetric matrices with an absolutely continuous distribution, we conclude that random matrices drawn from this ensemble will almost surely have simple eigenvalues.
For discrete random matrix ensembles, though, the above argument breaks down, even though general universality heuristics predict that the statistics of discrete ensembles should behave similarly to those of continuous ensembles. A model case here is the adjacency matrix Image may be NSFW.
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of an Erdös-Rényi graph – a graph on Image may be NSFW.
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of being in the graph. For the purposes of this paper one should view Image may be NSFW.
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, while Image may be NSFW.
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Theorem 1 With probability Image may be NSFW.
Clik here to view., Image may be NSFW.
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Our argument works for more general Wigner-type matrix ensembles, but for sake of illustration we will stick with the Erdös-Renyi case. Previous work on local universality for such matrix models (e.g. the work of Erdos, Knowles, Yau, and Yin) was able to show that any individual eigenvalue gap Image may be NSFW.
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did not vanish with probability Image may be NSFW.
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(in fact Image may be NSFW.
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for some absolute constant Image may be NSFW.
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), but because there are Image may be NSFW.
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Our argument in fact gives simplicity of the spectrum with probability Image may be NSFW.
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for any fixed Image may be NSFW.
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; in a subsequent paper we also show that it gives a quantitative lower bound on the eigenvalue gaps (analogous to how many results on the singularity probability of random matrices can be upgraded to a bound on the least singular value).
The basic idea of argument can be sketched as follows. Suppose that Image may be NSFW.
Clik here to view. has a repeated eigenvalue Image may be NSFW.
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. We split
Image may be NSFW.
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for a random Image may be NSFW.
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minor Image may be NSFW.
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and a random sign vector Image may be NSFW.
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; crucially, Image may be NSFW.
Clik here to view. and Image may be NSFW.
Clik here to view. are independent. If Image may be NSFW.
Clik here to view. has a repeated eigenvalue Image may be NSFW.
Clik here to view., then by the Cauchy interlacing law, Image may be NSFW.
Clik here to view. also has an eigenvalue Image may be NSFW.
Clik here to view.. We now write down the eigenvector equation for Image may be NSFW.
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Image may be NSFW.
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Extracting the top Image may be NSFW.
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coefficients, we obtain
Image may be NSFW.
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If we let Image may be NSFW.
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be the Image may be NSFW.
Clik here to view.-eigenvector of Image may be NSFW.
Clik here to view., then by taking inner products with Image may be NSFW.
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Image may be NSFW.
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we typically expect Image may be NSFW.
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to be non-zero, in which case we arrive at
Image may be NSFW.
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In other words, in order for Image may be NSFW.
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Clik here to view. has to be orthogonal to an eigenvector Image may be NSFW.
Clik here to view. of the minor Image may be NSFW.
Clik here to view.. Note that Image may be NSFW.
Clik here to view. and Image may be NSFW.
Clik here to view. are going to be independent (once we specify which eigenvector of Image may be NSFW.
Clik here to view. to take as Image may be NSFW.
Clik here to view.). On the other hand, thanks to inverse Littlewood-Offord theory (specifically, we use an inverse Littlewood-Offord theorem of Nguyen and Vu), we know that the vector Image may be NSFW.
Clik here to view. is unlikely to be orthogonal to any given vector Image may be NSFW.
Clik here to view. independent of Image may be NSFW.
Clik here to view., unless the coefficients of Image may be NSFW.
Clik here to view. are extremely special (specifically, that most of them lie in a generalised arithmetic progression). The main remaining difficulty is then to show that eigenvectors of a random matrix are typically not of this special form, and this relies on a conditioning argument originally used by Komlós to bound the singularity probability of a random sign matrix. (Basically, if an eigenvector has this special form, then one can use a fraction of the rows and columns of the random matrix to determine the eigenvector completely, while still preserving enough randomness in the remaining portion of the matrix so that this vector will in fact not be an eigenvector with high probability.)