Singular Values Distribution of Squares of Elliptic Random Matrices and Type B Narayana Polynomials

Abstract

We consider Gaussian elliptic random matrices X of a size N × N with parameter ρ, i.e., matrices whose pairs of entries (X i j , X ji ) are mutually independent Gaussian vectors with E X i j = 0, E X 2 i j = 1 and E X i j X ji = ρ. We are interested in the asymptotic distribution of eigenvalues of the matrix W = 1 N 2 X 2 X * 2 . We show that this distribution is determined by its moments, and we provide a recurrence relation for these moments. We prove that the (symmetrized) asymptotic distribution is determined by its free cumulants, which are Narayana polynomials of type B: c 2n = n k=0 n k 2 ρ 2k .

Universal hypotrochoidic law for random matrices with cyclic correlations

“…with t being the same as in Eq. (12). We note that for larged, t ∼d −1/2 , so that we recover the support (11) for full matrices with an effective parameter ρ 3 ∼d −1/2 .…”

“…For instance, the effects of negative correlations, which in systems theory corresponds to a rotation of the poles, is also explained by Eq. (11) and (12). On the other hand, the combination of cycles with positive and negative feedbacks for the same length is not fully covered by our method: in the case of a full matrix or a dense digraph, the positive and negative feedbacks appear in a single connected graph and thus cancel each other -meaning that Eq.…”

“…A key result in this field is the elliptic law [7], which states that in the limit of large matrix size the eigenvalues for random matrices with correlations between symmetric pairs of entries are confined within an ellipse in the complex plane. This result originating from over 30 years ago still drives scientific developments today, in both mathematical theory [8][9][10][11][12] as well as applications [5,13].…”

“…For a direct comparison we rescale the adjacency matrix of the graph byd −1/2 , which corresponds to the factor σ = N −1/2 that we used before and that normalizes rows and columns. The previous result (12) for k = 3 then reads…”

“…On the other hand, the combination of cycles with positive and negative feedbacks for the same length is not fully covered by our method: in the case of a full matrix or a dense digraph, the positive and negative feedbacks appear in a single connected graph and thus cancel each other -meaning that Eq. (11) and (12) remain valid -, but in very sparse digraphs the presence of many finite-size isolated sub-graphs implies that there are components dominated by either positive or negative feedback, and thus the effective medium approximation does not hold.…”