---
abstract: |
    For fixed $m > 1$, we study the product of $m$ independent $N \times N$ elliptic random matrices as $N$ tends to infinity. Our main result shows that the empirical spectral distribution of the product converges, with probability $1$, to the $m$-th power of the circular law, regardless of the joint distribution of the mirror entries in each matrix. This leads to a new kind of universality phenomenon: the limit law for the product of independent random matrices is independent of the limit laws for the individual matrices themselves.

    Our result also generalizes earlier results of Götze–Tikhomirov [@GTprod] and O’Rourke–Soshnikov [@OS] concerning the product of independent iid random matrices.
address:
- 'Department of Mathematics, University of Colorado at Boulder, Boulder, CO 80309 '
- 'Department of Mathematics, UCLA '
- 'Department of Mathematics, University of California, Davis, One Shields Avenue, Davis, CA 95616-8633 '
- 'Department of Mathematics, Yale University, New Haven, CT 06520, USA'
author:
- 'Sean O’Rourke'
- David Renfrew
- Alexander Soshnikov
- Van Vu
title: Products of independent elliptic random matrices
---

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Introduction
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We begin by recalling that the *eigenvalues* of a $N \times N$ matrix ${\mathbf}{M}$ are 