Laws of large numbers for eigenvectors and eigenvalues associated to random subspaces in a tensor product

Details Laws of large numbers for eigenvectors and eigenvalues associated to random subspaces in a tensor product Laws of large numbers for eigenvectors and eigenvalues associated to random subspaces in a tensor product

2010-08-18

arxiv.org/abs/1008.3099v3

Mathematics

Probability

v3 changes: minor typographic improvements; accepted version

Scientific paper

Given two positive integers $n$ and $k$ and a parameter $t\in (0,1)$, we choose at random a vector subspace $V_{n}\subset \mathbb{C}^{k}\otimes\mathbb{C}^{n}$ of dimension $N\sim tnk$. We show that the set of $k$-tuples of singular values of all unit vectors in $V_n$ fills asymptotically (as $n$ tends to infinity) a deterministic convex set $K_{k,t}$ that we describe using a new norm in $\R^k$. Our proof relies on free probability, random matrix theory, complex analysis and matrix analysis techniques. The main result result comes together with a law of large numbers for the singular value decomposition of the eigenvectors corresponding to large eigenvalues of a random truncation of a matrix with high eigenvalue degeneracy.

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