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 extension to d-dimensional phase distributions. Upper and lower bound asymptotics for the maximum singular value are found to be O(logN) and O(logN/ log logN) respectively where N is the dimension of the matrix, generalizing the results in [21]. We further study the behavior of the minimum singular value of a random Vandermonde matrix. In particular, we prove that the minimum singular value is at most N exp(−C √ 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 obtained 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 lower bound for the maximum absolute value of a random complex polynomial on the unit circle. We believe that this has independent mathematical interest. Lastly, for each sequence of positive integers {kp}p=1 we present a generalized version of the previously discussed random Vandermonde matrices. The classical random Vandermonde matrix corresponds to the sequence kp = 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,∞). Finally, we show that for the sequence kp = 2 p the limit eigenvalue distribution is the famous Marchenko–Pastur distribution.