Eigenvalue Results for Large Scale Random Vandermonde Matrices with Unit Complex Entries

This paper centers on the limit eigenvalue distribution for random Vandermonde matrices with unit magnitude complex entries. The phases of the entries are chosen independently and identically distributed from the interval $[-\pi,\pi]$. Various types of distribution for the phase are considered and we establish the existence of the empirical eigenvalue distribution in the large matrix limit on a wide range of cases. The rate of growth of the maximum eigenvalue is examined and shown to be no greater than $O(\log N)$ and no slower than $O(\log N/\log\log N)$ where $N$ is the dimension of the matrix. Additional results include the existence of the capacity of the Vandermonde channel (limit integral for the expected log determinant).