Distribution of Eigenvalues of Weighted, Structured Matrix Ensembles

Given a structured random matrix ensemble where each random variable occurs o(N) times in each row and the limiting rescaled spectral measure e μ exists, we fix a p 2 [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 ✏ij 2 {1, 1}, with Prob(✏ij = 1) = p (i.e., the Hadamard product). For p = 1/2 the limiting signed rescaled spectral measure is the semi-circle; for other p it has bounded (resp., unbounded) support if e μ has bounded (resp., unbounded) support, and converges to e μ as p ! 1. The proofs are by Markov’s Method of Moments, and involve 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 (the Catalan numbers). We discover and prove similar formulas for other configurations. 1We thank Colin Adams, Arup Bose, Satyan Devadoss, Allison Henrich, Murat Koloǧlu and Gene Kopp for helpful conversations, and the referee for many valuable suggestions. The first and third named authors were partially supported by NSF grant DMS0850577 and Williams College; the second, fourth and fifth named authors were partially supported by NSF grant DMS0970067, and the third author was also partially supported by NSF grant DMS1265673. INTEGERS: 15 (2015) 2