Abstract. The Gibbs distribution for n particles of the Coulomb gas on the unit circle at inverse temperature β is given by E β n(f) = 1 · · · f(e

Abstract. We present new algorithms that efficiently approximate the hypergeometric function of a matrix argument through its expansion as a series of Jack functions. Our algorithms exploit the combinatorial properties of the Jack function, and have complexity that is only linear in the size of the matrix. 1.

Abstract. We discuss a number of properties of CMV matrices, by which we mean the class of unitary matrices recently introduced by Cantero, Moral, and Velazquez. We argue that they play an equivalent role among unitary matrices to that of Jacobi matrices among all Hermitian matrices. In particular, we describe the analogues of well-known properties of Jacobi matrices: foliation by co-adjoint orbits, a natural symplectic structure, algorithmic reduction to this shape, Lax representation for an integrable lattice system (Ablowitz-Ladik), and the relation to orthogonal polynomials. As offshoots of our analysis, we will construct action/angle variables for the finite Ablowitz-Ladik hierarchy and describe the long-time behaviour of this system. 1.

Abstract not found

We derive Painlevé–type expressions for the distribution of the m th largest eigenvalue in the Gaussian Orthogonal and Symplectic Ensembles in the edge scaling limit. The work of Johnstone and Soshnikov (see [7], [10]) implies the immediate relevance of our formulas for the m th largest eigenvalue of the appropriate Wishart distribution. 1

We prove that the largest eigenvalues of the beta ensembles of random matrix theory converge in distribution to the low-lying eigenvalues of the random Schrödinger operator − d2 dx 2 + x + 2 √ β b ′ x restricted to the positive half-line, where b ′ x is white noise. In doing so we extend the definition of the Tracy-Widom(β) distributions to all β> 0, and also analyze their tails. Last, in a parallel development, we provide a second characterization of these laws in terms of a one-dimensional diffusion. The proofs rely on the associated tridiagonal matrix models and a universality result showing that the spectrum of such models converge to that of their continuum operator limit. In particular, we show how Tracy-Widom laws arise from a functional central limit theorem. 1

Our interest is in the cumulative probabilities Pr(L(t) ≤ l) for the maximum length of increasing subsequences in Poissonized ensembles of random permutations, random fixed point free involutions and reversed random fixed point free involutions. It is shown that these probabilities are equal to the hard edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively. The gap probabilities can be written as a sum over correlations for certain determinantal point processes. From these expressions a proof can be given that the limiting form of Pr(L(t) ≤ l) in the three cases is equal to the soft edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively, thereby reclaiming theorems due to Baik-Deift-Johansson and Baik-Rains. 1

Abstract. Let κ be the condition number of an m-by-n matrix with independent standard Gaussian entries, either real (β = 1) or complex (β = 2). The major result is the existence of a constant C (depending on m, n, β) such that P [κ> x] < C x −β for all x. As x → ∞, the bound is asymptotically tight. An analytic expression is given for the constant C, and simple estimates are given, one involving a Tracy-Widom largest eigenvalue distribution. All of the results extend beyond real and complex entries to general β.

Abstract. We provide a solution to the β-Jacobi matrix model problem posed by Dumitriu and the first author. The random matrix distribution introduced here, called a matrix model, is related to the model of Killip and Nenciu, but the development is quite different. We start by introducing a new matrix decomposition and an algorithm for computing this decomposition. Then we run the algorithm on a Haar-distributed random matrix to produce the β-Jacobi matrix model. The Jacobi ensemble on R n, parameterized by β> 0, a> −1, and b> −1, is the probability distribution whose density is proportional to Q β 2 i λ (a+1)−1

ensembles via matrix models