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Suppose I sample $m\times n$ matrix $X$ with IID entries sampled from standard Gaussian and $m,n$ are both large. What is the distribution of $k$th squared singular value?

I simulated singular values from some 100x100 random matrices below. Means of these seem to be well predicted by Marchenko-Pastur distribution with shape parameter 1. What about the variances?

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Notebook

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Looks like for the top eigenvalue, distribution approaches Tracy-Widom (notebook)

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