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 structured 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 ensemble 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 ǫ ij ∈ {1, −1}, with Prob(ǫ ij = 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.The proofs are by Markov's Method of Moments. The analysis of the 2k th moment for such distributions involves the pairings of 2k vertices on a circle. The contribution of each pairing in the signed case is weighted by a factor depending on p and the number of vertices involved in at least one crossing. These numbers are of interest in their own right, appearing in problems in combinatorics and knot theory. The number of configurations with no vertices involved in a crossing is well-studied, and are the Catalan numbers. We discover and prove similar formulas for configurations with 4, 6, 8 and 10 vertices in at least one crossing. We derive a closed-form expression for the expected value and determine the asymptotics for the variance for the number of vertices in at least one crossing. As the variance converges to 4, these results allow us to deduce properties of the limiting measure.

Bulk behavior of Schur–Hadamard products of symmetric random matrices

“…Thus, for example, a randomly (±1) censored Toeplitz matrix will have the semi-circular law as its LSD. This is the result of Beckwith et al [2011] in the p = 0.5 case.…”

“…In this article we shall consider Schur-Hadamard products of real symmetric patterned matrices and establish results on their LSD. In particular, we prove an invariance theorem which yields the result of Beckwith et al [2011], when p = 0.5, as a special case. We also consider the Schur-Hadamard product of Toeplitz and Hankel matrices (and other combinations like Toeplitz and Reverse Circulant etc.)…”

“…More recently, Beckwith et al [2011] considered Schur-Hadamard product of a ±1 Bernoulli(p) Wigner matrix with a Toeplitz matrix and established the existence of the LSD. They found that when p = 0.5, the LSD is the familiar semi-circular law, but when p = 0.5, the limiting moments are polynomials in (2p − 1) whose coefficients could not be identified, and as p approaches 1 these moments approach the corresponding moments of the LSD of the Toeplitz matrix.…”