Asymptotic behavior of the maximum and minimum singular value of random vandermonde matrices

This study 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 Ω((log log N^d/(log log N^d))^1/2, respectively, where N is the dimension of the matrix, generalizing the results in Tucci and Whiting (IEEE Trans Inf Theory 57(6):3938-3954, 2011). 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√N)) with high probability where C is a constant independent of 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 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 show that the limit eigenvalue distribution converges to a probability measure supported on [0, ∞). Finally, we show that for the sequence k_p = 2_p the limit eigenvalue distribution is the famous Marchenko-Pastur distribution.