# Wangjun Yuan: On spectral distribution of sample covariance matrices from large dimensional and large $k$-fold tensor products

**Time: **
Tue 2023-04-04 13.15 - 14.15

**Location: **
KTH, room 3418

**Participating: **
Wangjun Yuan

**Abstract.**

We study the eigenvalue distributions for sums of independent rank-one *k*-fold tensor products of large *n*-dimensional vectors. Previous results in the literature assume that *k=o(n)* and show that the eigenvalue distributions converge to the celebrated Mar\v{c}enko-Pastur law under appropriate moment conditions on the base vectors. In this paper, motivated by quantum information theory, we study the regime where *k* grows faster, namely *k=O(n)*. We show that the moment sequences of the eigenvalue distributions have a limit, which is different from the Marcenko-Pastur law, and the Mar\v{c}enko-Pastur law limit holds if and only if *k=o(n) *for this tensor model. The approach is based on the method of moments.

This is a joint work with Benoit Collins and Jianfeng Yao.