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

TL;DR

This paper investigates the asymptotic behavior of the largest and smallest singular values of random Vandermonde matrices, extending to multi-dimensional phase distributions and providing bounds and explicit constants.

Contribution

It introduces new asymptotic bounds for singular values of random Vandermonde matrices and explores their eigenvalue distributions, including cases with generalized sequences.

Findings

Maximum singular value grows as O(√log N^d)

Minimum singular value is at most N exp(-C√N) with high probability

Eigenvalue distribution converges to Marchenko--Pastur law for specific sequences

Abstract

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 $O(\log^{1/2}{N^{d}})$ and $\Omega((\log N^{d} /(\log \log N^d))^{1/2})$ respectively where $N$ is the dimension of the matrix, generalizing the results in \cite{TW}. 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 on $N$. 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 we 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.