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.