Asymptotic Results on Genralized Vandermonde Matrices and their Extreme Eigenvalues

This work examines various statistical distributions in connection with random Vandermonde matrices and their generalization to $d$-dimensional phase distributions. Upper and lower bound asymptotics for the maximum eigenvalue are found to be $O(log N^d)$ and $O(log N^{d} /log log N^d)$ respectively, generalizing the results in cite{TW}. The behavior of the minimum eigenvalue is considered by studying the behavior of the maximum eigenvalue of the inverse matrix. In particular, we prove that the minimum eigenvalue $lambda_1$ is shown to be at most $O(exp(-sqrt{N}W_{N}^{*}))$ where $W_N^*$ is a positive random variable converging weakly to a random variable constructed from a realization of the Brownian Bridge on $[0,2pi)$. Additional results for $lb {bf V}^* {bf V} rb^{-1}$, a trace log formula for ${bf V}^* {bf V}$, as well as a some numerical examinations of the size of the atom at $0$ for the random Vandermonde eigenvalue distribution are also presented.