Narrow Results

We study the joint distribution of the set of all marginals of a random Wishart matrix acting on a tensor product Hilbert space. We compute the limiting free mixed cumulants of the marginals, and we show that in the balanced asymptotical regime, the marginals are asymptotically free. We connect the matrix integrals relevant to the study of operators on tensor product spaces with the corresponding classes of combinatorial maps, for which we develop the combinatorial machinery necessary for the asymptotic study. Finally, we present some applications to the theory of random quantum states in quantum information theory.

We prove the Marchenko–Pastur law for the eigenvalues of $p\times p$ sample covariance matrices in two new situations where the data does not have independent coordinates. In the first scenario — the block-independent model — the $p$ coordinates of the data are partitioned into blocks in such a way that the entries in different blocks are independent, but the entries from the same block may be dependent. In the second scenario — the random tensor model — the data is the homogeneous random tensor of order $d$, i.e. the coordinates of the data are all $\left(\genfrac{}{}{0.0pt}{}{n}{d}\right)$ different products of $d$ variables chosen from a set of $n$ independent random variables. We show that Marchenko–Pastur law holds for the block-independent model as long as the size of the largest block is $o(p)$, and for the random tensor model as long as $d=o(n^{1/3})$. Our main technical tools are new concentration inequalities for quadratic forms in random variables with block-independent coordinates, and for random tensors.

Suppose that $y_{\alpha }^{(i)},i=1,\dots ,k,\alpha =1,\dots ,m$ are i.i.d. copies of random vector $y={(y_1,y_2,\dots ,y_p)}^T$. Let

In this paper, we consider an initial $n\times d$ random matrix with non-Gaussian correlated entries on each row and independent entries from one row to another. The correlation on the rows is given by the correlation of the increments of the Rosenblatt process, which is a non-Gaussian self-similar process with stationary increments, living in the second Wiener chaos. To this initial matrix, we associate a Wishart tensor of length $p\ge 2$. We study the limit behavior in distribution of this Wishart tensor in the high-dimensional regime, i.e. when $n,d$ are large enough. We prove that the vector corresponding to the $p$-Wishart tensor converges to an $n^p$-dimensional non-Gaussian vector, with Rosenblatt random variables on its hyper-diagonals and zeros outside the hyper-diagonals.