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 \(\Omega ((\log N^{d} /(\log \log N^d))^{1/2})\), 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.