Plancherel–rotach Formulae for Average Characteristic Polynomials of Products of Ginibre Random Matrices and the Fuss–catalan Distribution

Abstract

Formulae of Plancherel-Rotach type are established for the average characteristic polynomials of certain Hermitian products of rectangular Ginibre random matrices on the region of zeros. These polynomials form a general class of multiple orthogonal hypergeometric polynomials generalizing the classical Laguerre polynomials. The proofs are based on a multivariate version of the complex method of saddle points. After suitable rescaling the asymptotic zero distributions for the polynomials are studied and shown to coincide with the Fuss-Catalan distributions. Moreover, introducing appropriate coordinates, elementary and explicit characterizations are derived for the densities as well as for the distribution functions of the Fuss-Catalan distributions of general order.

Singular Values for Products of Complex Ginibre Matrices with a Source: Hard Edge Limit and Phase Transition

“…The first point to note is that the hard edge state in the case A = 0 depends on r, and thus is no longer described by the correlation kernel (1.3). This fact can be anticipated by an analysis of the global density of the squared singular values [5,31,53,56]. The global density, which refers to the limiting density of eigenvalues of N −r−1 Y * Y as N → ∞, is found to exhibit the hard edge singularity (see [56] or [31, eqn (2.16)…”

Asymptotic zero distribution of Jacobi-Piñeiro and multiple Laguerre polynomials

“…· · · (k + ν r )! ,and the asymptotic distribution of the scaled zeros {x k,n /n r , 1 ≤ k ≤ n} is given in[16, Thm. 3.2].…”

Spectral densities of singular values of products of Gaussian and truncated unitary random matrices

“…Using properties of the S-transform from free probability theory it can be derived (see, e.g., [13]) that the function w(z) = zF (z) satisfies the algebraic equation Up to a scaling in the argument, this is the equation for the Stieltjes transforms in the case s = 0, which coincides with the Fuss-Catalan case. It is known (see, e.g., [11,21]) that the boundary values of v on the branch cut (0, x * ) can be stated explicitly by This equation is of a similar type as (3.8), which enables us to find a functional relation between G and the Stieltjes transform F of µ r,r in terms of a rational transformation F (z) = (r + 1)2 r G 2 r+1 z 1 + (r − 1)2 r zG (2 r+1 z) .…”

“…However, so far the densities of µ r,s are known only in the special cases s = 0, s = 1 and s = r. In the case s = 0 the product (1.1) consists only of complex Gaussian matrices and the limiting distribution of eigenvalues of (1.4) is given by the Fuss-Catalan distribution of order r. The corresponding density can be expressed in terms of elementary functions [5], [21] by…”