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

Abstract

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 $\Omega ((\log N^{d} /(\log \log N^d))^{1/2})$, 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 $N\exp (-C\sqrt{N}))$ 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 $\{k_p\}_{p=1}^{\infty }$ 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 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.