Distribution of Eigenvalues of Weighted, Structured Matrix Ensembles
Abstract
The limiting distribution of eigenvalues of N x N random matrices has many applications. One of the most studied ensembles are real symmetric matrices with independent entries iidrv; the limiting rescaled spectral measure (LRSM) $\widetilde{\mu}$ is the semicircle. Studies have determined the LRSMs for many structured ensembles, such as Toeplitz and circulant matrices. These have very different behavior; the LRSM for both have unbounded support. Given a structured ensemble such that (i) each random variable occurs o(N) times in each row and (ii) the LRSM exists, we introduce a parameter to continuously interpolate between these behaviors. We fix a p in [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 epsilon_ij in {1, 1}, with Prob(epsilon_ij = 1) = p (i.e., the Hadamard product). For p = 1/2 we prove that the limiting signed rescaled spectral measure is the semicircle. For all other p, the limiting measure has bounded (resp., unbounded) support if $\widetilde{\mu}$ has bounded (resp., unbounded) support, and converges to $\widetilde{\mu}$ as p > 1. Notably, these results hold for Toeplitz and circulant matrix ensembles. The proofs are by the Method of Moments. The analysis involves the pairings of 2k vertices on a circle. The contribution of each 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 appear in combinatorics and knot theory. The number of configurations with no vertices involved in a crossing is wellstudied, and are the Catalan numbers. We prove similar formulas for configurations with up to 10 vertices in at least one crossing. We derive a closedform expression for the expected value and determine the asymptotics for the variance for the number of vertices in at least one crossing.
 Publication:

arXiv eprints
 Pub Date:
 December 2011
 arXiv:
 arXiv:1112.3719
 Bibcode:
 2011arXiv1112.3719B
 Keywords:

 Mathematics  Probability;
 15B52;
 60F05;
 11D45 (primary);
 60F15;
 60G57;
 62E20 (secondary)
 EPrint:
 Version 3.0, 29 pages, 3 figures. Added clarification on the applicability of the main theorem (ie, what ensembles may be studied)