Asymptotic Behavior of the Maximum and Minimum Singular Value of Random Vandermonde Matrices

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 singular value are found to be $O(log N^d)$ and $O(log N^{d} /log log N^d)$ 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 a random Vandermonde matrix. In particular, we prove that the minimum singular value $lambda_1$ is at most $N^2exp(-Csqrt{N}))$ where $N$ is the dimension of the matrix and $C$ is a constant. Furthermore, the value of the constant $C$ is determined explicitly. The main result is obtain in two different ways. One approach uses techniques from stochastic processes and in particular, a construction related with the Brownian bridge. The other one is a more direct analytical approach involving combinatorics and complex analysis. As a consequence, we obtain a result on the lower bound of the maximum of a random polynomial on the unit circle. We believe that this has independent mathematical interest.