The Ginibre point process (GPP) is one of the main examples of determinantal point processes on the complex plane. It is a recurring distribution of random matrix theory as well as a useful model in applied mathematics. In this paper we briefly overview the usual methods for the simulation of the GPP. Then we introduce a modified version of the GPP which constitutes a determinantal point process more suited for certain applications, and we detail its simulation. This modified GPP has the property of having a fixed number of points and having its support on a compact subset of the plane. See Decreusefond et al. (2013) for an extended version of this paper.

We introduce and study a remarkable family of real probability measures ${{\pi }_{st}}$ that we call free Bessel laws. These are related to the free Poisson law $\pi $ via the formulae ${{\text{ }\!\!\pi\!\!\text{ }}_{s1}}={{\text{ }\!\!\pi\!\!\text{ }}^{\boxtimes s}}$ and $\text{ }\pi {{\text{ }}_{1t}}=\text{ }\pi {{\text{ }}^{\boxtimes }}^{t}$. Our study includes definition and basic properties, analytic aspects (supports, atoms, densities), combinatorial aspects (functional transforms, moments, partitions), and a discussion of the relation with random matrices and quantum groups.

We study the geometry, topology and Lebesgue measure of the set of monic conjugate reciprocal polynomials of fixed degree with all roots on the unit circle. The set of such polynomials of degree $N$ is naturally associated to a subset of ${{\mathbb{R}}^{N-1}}$. We calculate the volume of this set, prove the set is homeomorphic to the $N-1$ ball and that its isometry group is isomorphic to the dihedral group of order $2N$.

We prove a self-adjoint analogue of the Marcus–Pisier inequality, comparing the expected value of convex functionals on random reflection matrices and on elements of the Gaussian orthogonal (or unitary) ensemble.

We give a new heuristic for all of the main terms in the integral moments of various families of primitive $L$-functions. The results agree with previous conjectures for the leading order terms. Our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary, orthogonal, or symplectic matrices, where the moments are defined by the appropriate group averages. This lends support to the idea that arithmetical $L$-functions have a spectral interpretation, and that their value distributions can be modelled using Random Matrix Theory. Numerical examples show good agreement with our conjectures.

Okamoto has obtained a sequence of τ-functions for the PVI system expressed as a double Wronskian determinant based on a solution of the Gauss hypergeometric equation. Starting with integral solutions of the Gauss hypergeometric equation, we show that the determinant can be re-expressed as multidimensional integrals, and these in turn can be identified with averages over the eigenvalue probability density function for the Jacobi unitary ensemble (JUE), and the Cauchy unitary ensemble (CyUE) (the latter being equivalent to the circular Jacobi unitary ensemble (cJUE)). Hence these averages, which depend on four continuous parameters and the discrete parameter N, can be characterised as the solution of the second order second degree equation satisfied by the Hamiltonian in the PVI theory. We show that the Hamiltonian also satisfies an equation related to the discrete PV equation, thus providing an alternative characterisation in terms of a difference equation. In the case of the cJUE, the spectrum singularity scaled limit is considered, and the evaluation of a certain four parameter average is given in terms of the general PV transcendent in σ form. Applications are given to the evaluation of the spacing distribution for the circular unitary ensemble (CUE) and its scaled counterpart, giving formulas more succinct than those known previously; to expressions for the hard edge gap probability in the scaled Laguerre orthogonal ensemble (LOE) (parameter a a non-negative integer) and Laguerre symplectic ensemble (LSE) (parameter a an even non-negative integer) as finite dimensional combinatorial integrals over the symplectic and orthogonal groups respectively; to the evaluation of the cumulative distribution function for the last passage time in certain models of directed percolation; to the τ-function evaluation of the largest eigenvalue in the finite LOE and LSE with parameter a = 0; and to the characterisation of the diagonal-diagonal spin-spin correlation in the two-dimensional Ising model.

Following Füredi and Komlós, we develop a graph theory method to study the high moments of large random matrices with independent entries. We apply this method to sparse N × N random matrices AN,p that have, on average, p non-zero elements per row. One of our results is related to the asymptotic behaviour of the spectral norm ∥AN,p∥ in the limit 1 ≪ p ≪ N. We show that the value pc = log N is the critical one for lim ∥AN,p/√p∥ to be bounded or not. We discuss relations of this result with the Erdős–Rényi limit theorem and properties of large random graphs. In the proof, the principal issue is that the averaged vertex degree of plane rooted trees of k edges remains bounded even when k → ∞. This observation implies fairly precise estimates for the moments of AN,p. They lead to certain generalizations of the results by Sinai and Soshnikov on the universality of local spectral statistics at the border of the limiting spectra of large random matrices.

The genus series for maps is the generating series for the number of rooted maps with a given number of vertices and faces of each degree, and a given number of edges. It captures topological information about surfaces, and appears in questions arising in statistical mechanics, topology, group rings, and certain aspects of free probability theory. An expression has been given previously for the genus series for maps in locally orientable surfaces in terms of zonal polynomials. The purpose of this paper is to derive an integral representation for the genus series. We then show how this can be used in conjunction with integration techniques to determine the genus series for monopoles in locally orientable surfaces. This complements the analogous result for monopoles in orientable surfaces previously obtained by Harer and Zagier. A conjecture, subsequently proved by Okounkov, is given for the evaluation of an expectation operator acting on the Jack symmetric function. It specialises to known results for Schur functions and zonal polynomials.

Let M be a random matrix chosen from Haar measure on the unitary group Un. Let Z = X + iY be a standard complex normal random variable with X and Y independent, mean 0 and variance ½ normal variables. We show that for j = 1, 2, …, Tr(Mj) are independent and distributed as √jZ asymptotically as n →∞. This result is used to study the set of eigenvalues of M. Similar results are given for the orthogonal and symplectic and symmetric groups.