DISTRIBUTION OF EIGENVALUES OF WEIGHTED, STRUCTURED MATRIX ENSEMBLES

ABSTRACT. The study of the limiting distribution of eigenvalues of N × N random matrices as N → ∞ has many applications, including nuclear physics, number theory and network theory. One of the most studied ensembles is that of real symmetric matrices with independent entries drawn from identically distributed nice random variables, where the limiting rescaled spectral measure is the semi-circle. Studies have also determined the limiting rescaled spectral measures for many struc-tured ensembles, such as Toeplitz and circulant matrices. These systems have very different behavior; the limiting rescaled spectral measures for both have unbounded support. Given a structured ensem-ble such that (i) each random variable occurs o(N) times in each row of matrices in the ensemble and (ii) the limiting rescaled spectral measure µ ̃ exists, we introduce a parameter to continuously interpolate between these two behaviors. We fix a p ∈ [1/2, 1] and study the ensemble of signed structured matrices by multiplying the (i, j)th and (j, i)th entries of a matrix by a randomly chosen ǫij ∈ {1,−1}, with Prob(ǫij = 1) = p (i.e., the Hadamard product). For p = 1/2 we prove that the limiting signed rescaled spectral measure is the semi-circle. For all other p, we prove the limiting measure has bounded (resp., unbounded) support if µ ̃ has bounded (resp., unbounded) support, and converges to µ ̃ as p → 1. Notably, these results hold for Toeplitz and circulant matrix ensembles