Empirical Distribution of Scaled Eigenvalues for Product of Matrices from the Spherical Ensemble

Consider the product of $m$ independent $n\times n$ random matrices from the spherical ensemble for $m\ge 1$. The empirical distribution based on the $n$ eigenvalues of the product is called the empirical spectral distribution. Two recent papers by G\"otze, K\"osters and Tikhomirov (2015) and Zeng (2016) obtain the limit of the empirical spectral distribution for the product when $m$ is a fixed integer. In this paper, we investigate the limiting empirical distribution of scaled eigenvalues for the product of $m$ independent matrices from the spherical ensemble in the case when $m$ changes with $n$, that is, $m=m_n$ is an arbitrary sequence of positive integers.