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Given two positive integers $n$ and $k$ and a parameter $t\in (0,1)$, wechoose at random a vector subspace $V_{n}\subset\mathbb{C}^{k}\otimes\mathbb{C}^{n}$ of dimension $N\sim tnk$. We show that theset of $k$-tuples of singular values of all unit vectors in $V_n$ fillsasymptotically (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 analysisand matrix analysis techniques. The main result result comes together with alaw of large numbers for the singular value decomposition of the eigenvectorscorresponding to large eigenvalues of a random truncation of a matrix with higheigenvalue degeneracy.

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