Triangular random matrices and biorthogonal ensembles

Abstract

We study the singular values of certain triangular random matrices. When their elements are i.i.d. standard complex Gaussian random variables, the squares of the singular values form a biorthogonal ensemble, and with an appropriate change in the distribution of the diagonal elements, they give the biorthogonal Laguerre ensemble. For triangular Wigner matrices, we give alternative proofs for the convergence of the empirical distribution of the appropriately scaled squares of the singular eigenvalues to a distribution with support [0, e], as well as for the almost sure convergence of the rescaled largest singular eigenvalue to √ e under the additional assumption of mean zero and finite fourth moment for the law of the matrix elements. n i=1 δ λ n,m i 1

Selberg integral theory and Muttalib–Borodin ensembles

“…A significant feature of these weights in subsequent analysis is that they are all even. It follows that, with a suitable identification of the parameters, the change of variables x 2 = y maps the weights (6) to the weights (2), and consequently from the full real line to the positive half-line.…”

“…This was first isolated in the work of Cheliotis [6] as the eigenvalue PDF of the random matrix Y † Y , where Y is the random upper triangular matrix specified below (20). Setting γ j = θ(j − 1) + c shows that the determinant can then be evaluated as a product since it is an example of a Vandermonde determinant, and (26) reduces to (1) with the Laguerre weight from (2), parameter a = c, as already noticed in [6]. In particular, after reinstating the parameter a, we have for the normalisation Laguerre Muttalib-Borodin ensemble…”

“…Recently, the PDFs (1) have drawn renewed attention in the random matrix literature [7,28,6,19,21,18] with the name "Muttalib-Borodin ensembles" coined in [21]. This increase in attention is justified by newly discovered links to several applications, for example to triangular random matrix models [6] and to certain combinatorial numbers [19]. Prominently, there also exists an intimate connection to products of random matrices and free probability when either θ or 1/θ is an integer [28].…”

“…We now turn our attention to calculating the normalisation for the Muttalib-Borodin ensembles supported on the full real line, i.e. PDFs (5) with the weights (6). The tractability of this task relies on exploiting an underlying parity symmetry.…”

Gaussian perturbations of hard edge random matrix ensembles

“…The last example consists of random matrices for which the joint probability density of eigenvalues is the Muttalib-Borodin Laguerre ensemble [9,26] 1 Such densities can be realized as eigenvalue densities of random matrices, see [1,11,19]. In [19], the authors constructed a random matrix with this eigenvalue density in the following way, in the case where θ is a positive integer and α a non-negative integer.…”