The paper has three parts. In the first part we apply the theory of commuting pairs of (pseudo) difference operators to the (formal) asymptotics of orthogonal polynomials: using purely geometrical arguments we show heuristically that the asymptotics, for large degrees, of orthogonal polynomial with respect to varying weights is intimately related to certain spinor bundles on a hyperelliptic algebraic curve reproducing formulae appearing in the works of Deift et al. on the subject.

We consider the two matrix model with an even quartic potential W (y) = y 4 /4 + αy 2 /2 and an even polynomial potential V (x). The main result of the paper is the formulation of a vector equilibrium problem for the limiting mean density for the eigenvalues of one of the matrices M 1 . The vector equilibrium problem is defined for three measures, with external fields on the first and third measures and an upper constraint on the second measure. The proof is based on a steepest descent analysis of a 4 × 4 matrix valued Riemann-Hilbert problem that characterizes the correlation kernel for the eigenvalues of M 1 . Our results generalize earlier results for the case α = 0, where the external field on the third measure was not present.

In this paper, we consider N -dimensional real Wishart matrices Y in the class W R (Σ, M ) in which all but one eigenvalues of Σ is 1. Let the non-trivial eigenvalue of Σ be 1 + τ , then as N , M → ∞, with M/N = γ 2 finite and non-zero, the eigenvalue distribution of Y will converge into the Marchenko-Pastur distribution inside a bulk region. When τ increases from zero, one starts to see a stray eigenvalue of Y outside of the support of the Marchenko-Pastur density. As the this stray eigenvalue leaves the bulk region, a phase transition will occur in the largest eigenvalue distribution of the Wishart matrix. In this paper we will compute the asymptotics of the largest eigenvalue distribution when the phase transition occur. We will first establish the results that are valid for all N and M and will use them to carry out the asymptotic analysis. In particular, we have derived a contour integral formula for the Harish-Chandra Itzykson-Zuber integral O(N ) e tr(XgY g T ) g T dg when X, Y are real symmetric and Y is a rank 1 matrix. This allows us to write down a Fredholm determinant formula for the largest eigenvalue distribution and analyze it using orthogonal polynomial techniques. As a result, we obtain an integral formula for the largest eigenvalue distribution in the large N limit characterized by Painlevé transcendents. The approach used in this paper is very different from a recent paper [23], in which the largest eigenvalue distribution was obtained using stochastic operator method. In particular, the Painlevé formula for the largest eigenvalue distribution obtained in this paper is new.

In this paper we study the Hamiltonian structure of the second Painlevé hierarchy, an infinite sequence of nonlinear ordinary differential equations containing PII as its simplest equation. The n-th element of the hierarchy is a non linear ODE of order 2n in the independent variable z depending on n parameters denoted by t 1 , . . . , t n−1 and αn. We introduce new canonical coordinates and obtain Hamiltonians for the z and t 1 , . . . , t n−1 evolutions. We give explicit formulae for these Hamiltonians showing that they are polynomials in our canonical coordinates.1991 Mathematics Subject Classification. Primary 34M55. Secondary 37K20, 35Q53.

We investigate the matrix model with weight w(x) := exp − z 2 2x 2 + * The authors acknowledge financial support by the EPSRC grant EP/G019843/1.