Abstract. Recent works have shown that the family of probability distributions with moments given by the Fuss-Catalan numbers permit a simple parameterized form for their density. We extend this result to the Raney distribution which by definition has its moments given by a generalization of the Fuss-Catalan numbers. Such computations begin with an algebraic equation satisfied by the Stieltjes transform, which we show can be derived from the linear differential equation satisfied by the characteristic polynomial of random matrix realizations of the Raney distribution. For the Fuss-Catalan distribution, an equilibrium problem characterizing the density is identified. The Stieltjes transform for the limiting spectral density of the singular values squared of the matrix product formed from q inverse standard Gaussian matrices, and s standard Gaussian matrices, is shown to satisfy a variant of the algebraic equation relating to the Raney distribution. Supported on (0, ∞), we show that it too permits a simple functional form upon the introduction of an appropriate choice of parameterization. As an application, the leading asymptotic form of the density as the endpoints of the support are approached is computed, and is shown to have some universal features.

It has been shown by Akemann, Ipsen and Kieburg that the squared singular values of products of M rectangular random matrices with independent complex Gaussian entries are distributed according to a determinantal point process with a correlation kernel that admits a representation in terms of Meijer G-functions. We prove the universality of the local statistics of the squared singular values, namely, the bulk universality given by the sine kernel and the edge universality given by the Airy kernel. The proof is based on the asymptotic analysis for the double contour integral representation of the correlation kernel. Our strategy can be generalized to deal with other models of products of random matrices introduced recently and to establish similar universal results. Two more examples are investigated, one is the product of M Ginibre matrices and the inverse of K Ginibre matrices studied by Forrester, and the other one is the product of M − 1 Ginibre matrices with one truncated unitary matrix considered by Kuijlaars and Stivigny.

In this paper we study entanglement of the reduced density matrix of a bipartite quantum system in a random pure state.It transpires that this involves the computation of the smallest eigenvalue distribution of the fixed trace Laguerre ensemble of N × N random matrices. We showed that for finite N the smallest eigenvalue distribution may be expressed in terms of Jack polynomials.Furthermore, based on the exact results, we found, a limiting distribution, when the smallest eigenvalue is suitably scaled with N followed by a large N limit. Our results turn out to be the same as the smallest eigenvalue distribution of the classical Laguerre ensembles without the fixed trace constraint. This suggests in a broad sense, the global constraint does not influence local correlations, at least, in the large N limit.Consequently, we have solved an open problem: The determination of the smallest eigenvalue distribution of the reduced density matrix-obtained by tracing out the environmental degrees of freedom-for a bipartite quantum system of unequal dimensions.

The singular values squared of the random matrix product Y = GrG r−1 · · · G 1 (G 0 + A), where each G j is a rectangular standard complex Gaussian matrix while A is non-random, are shown to be a determinantal point process with correlation kernel given by a double contour integral. When all but finitely many eigenvalues of A * A are equal to bN , the kernel is shown to admit a well-defined hard edge scaling, in which case a critical value is established and a phase transition phenomenon is observed. More specifically, the limiting kernel in the subcritical regime of 0 < b < 1 is independent of b, and is in fact the same as that known for the case b = 0 due to Kuijlaars and Zhang. The critical regime of b = 1 allows for a double scaling limit by choosing b = (1 − τ / √ N ) −1 , and for this the critical kernel and outlier phenomenon are established. In the simplest case r = 0, which is closely related to nonintersecting squared Bessel paths, a distribution corresponding to the finite shifted mean LUE is proven to be the scaling limit in the supercritical regime of b > 1 with two distinct scaling rates. Similar results also hold true for the random matrix product TrT r−1 · · · T 1 (G 0 +A), with each T j being a truncated unitary matrix.

We study the local properties of eigenvalues for the Hermite (Gaussian), Laguerre (Chiral) and Jacobi β-ensembles of N × N random matrices. More specifically, we calculate scaling limits of the expectation value of products of characteristic polynomials as N → ∞. In the bulk of the spectrum of each β-ensemble, the same scaling limit is found to be e p 1 1 F 1 whose exact expansion in terms of Jack polynomials is well known. The scaling limit at the soft edge of the spectrum for the Hermite and Laguerre β-ensembles is shown to be a multivariate Airy function, which is defined as a generalized Kontsevich integral. As corollaries, when β is even, scaling limits of the k-point correlation functions for the three ensembles are obtained. The asymptotics of the multivariate Airy function for large and small arguments is also given. All the asymptotic results rely on a generalization of Watson's lemma and the steepest descent method for integrals of Selberg type. 19

Abstract. Consider the product GX of two rectangular complex random matrices coupled by a constant matrix Ω, where G can be thought to be a Gaussian matrix and X is a bi-invariant polynomial ensemble. We prove that the squared singular values form a biorthogonal ensemble in Borodin's sense, and further that for X being Gaussian the correlation kernel can be expressed as a double contour integral. When all but finitely many eigenvalues of ΩΩ * are equal, the corresponding correlation kernel is shown to admit a phase transition phenomenon at the hard edge in four different regimes as the coupling matrix changes. Specifically, the four limiting kernels in turn are the Meijer G-kernel for products of two independent Gaussian matrices, a new critical and interpolating kernel, the perturbed Bessel kernel and the finite coupled product kernel associated with GX. In the special case that X is also a Gaussian matrix and Ω is scalar, such a product has been recently investigated by Akemann and Strahov. We also propose a Jacobi-type product and prove the same transition.

No abstract

We solve the problem on local statistics of finite Lyapunov exponents for M products of N ×N Gaussian random matrices as both M and N go to infinity, proposed by Akemann, Burda, Kieburg [3] and Deift [18]. When the ratio (M + 1)/N changes from 0 to ∞, we prove that the local statistics undergoes a transition from GUE to Gaussian. Especially at the critical scaling (M + 1)/N → γ ∈ (0, ∞), we observe a phase transition phenomenon.