  This work examines various statistical distributions in connection with
random Vandermonde matrices and their extension to $d$--dimensional phase
distributions. Upper and lower bound asymptotics for the maximum singular value
are found to be $O(\log^{1/2}{N^{d}})$ and $\Omega((\log N^{d} /(\log \log
N^d))^{1/2})$ respectively where $N$ is the dimension of the matrix,
generalizing the results in \cite{TW}. We further study the behavior of the
minimum singular value of these random matrices. In particular, we prove that
the minimum singular value is at most $N\exp(-C\sqrt{N}))$ with high
probability where $C$ is a constant independent on $N$. Furthermore, the value
of the constant $C$ is determined explicitly. The main result is obtained in
two different ways. One approach uses techniques from stochastic processes and
in particular, a construction related to the Brownian bridge. The other one is
a more direct analytical approach involving combinatorics and complex analysis.
As a consequence, we obtain a lower bound for the maximum absolute value of a
random complex polynomial on the unit circle, which may be of independent
mathematical interest. Lastly, for each sequence of positive integers
${k_p}_{p=1}^{\infty}$ we present a generalized version of the previously
discussed matrices. The classical random Vandermonde matrix corresponds to the
sequence $k_{p}=p-1$. We find a combinatorial formula for their moments and we
show that the limit eigenvalue distribution converges to a probability measure
supported on $[0,\infty)$. Finally, we show that for the sequence $k_p=2^{p}$
the limit eigenvalue distribution is the famous Marchenko--Pastur distribution.
