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Universality and least singular values of random matrix products: a simplified approach

In this note, we show how to provide sharp control on the least singular value of a certain translated linearization matrix arising in the study of the local universality of products of independent random matrices. This problem was first considered in a recent work of Koppel, O'Rourke, and Vu, and compared to their work, our proof is substantially simpler and established in much greater generality . In particular, we only assume that the entries of the ensemble are centered, and have second and fourth moments uniformly bounded away from $0$ and infinity, whereas previous work assumed a uniform subgaussian decay condition and that the entries within each factor of the product are identically distributed. A consequence of our least singular value bound is that the four moment matching universality results for the products of independent random matrices, recently obtained by Koppel, O'Rourke, and Vu, hold under much weaker hypotheses. Our proof technique is also of independent interest in the study of structured sparse matrices.