We study a model of n one-dimensional non-intersecting Brownian motions with two prescribed starting points at time t = 0 and two prescribed ending points at time t = 1 in a critical regime where the paths fill two tangent ellipses in the time-space plane as n → ∞. The limiting mean density for the positions of the Brownian paths at the time of tangency consists of two touching semicircles, possibly of different sizes. We show that in an appropriate double scaling limit, there is a new familiy of limiting determinantal point processes with integrable correlation kernels that are expressed in terms of a new RiemannHilbert problem of size 4 × 4. We prove solvability of the Riemann-Hilbert problem and establish a remarkable connection with the Hastings-McLeod solution of the Painlevé II equation. We show that this Painlevé II transcendent also appears in the critical limits of the recurrence coefficients of the multiple Hermite polynomials that are associated with the non-intersecting Brownian motions. Universality suggests that the new limiting kernels apply to more general situations whenever a limiting mean density vanishes according to two touching square roots, which represents a new universality class.

We consider the random matrix model with external source in the case where the potential V .x/ is an even polynomial and the external source has two eigenvalueṡ a of equal multiplicity. We show that the limiting mean eigenvalue distribution of this model can be characterized as the first component of a pair of measures . 1 ; 2 / that solve a constrained vector equilibrium problem. The proof is based on the steepest-descent analysis of the associated Riemann-Hilbert problem for multiple orthogonal polynomials.We illustrate our results in detail for the case of a quartic double-well potential V .x/ D 1 4 x 4 t 2 x 2 . We are able to determine the precise location of the phase transitions in the ta-plane, where either the constraint becomes active or the two intervals in the support come together (or both).

In this paper we introduce a Givens-weight representation for rank structured matrices, where the rank structure is defined by certain low rank submatrices starting from the bottom left matrix corner. This representation will be compared to the (block) quasiseparable representations occurring in the literature. We will then provide some basic algorithms for the Givens-weight representation, in particular showing how to obtain a Givens-weight representation for a full matrix, and how to reduce the order of the representation, whenever appropriate. We will also show how to update the representation under the action of Givens transformations, and how to compute matrix-vector products. As such, these results will be the basis for the algorithms on Givens-weight representations to be described in subsequent papers.

We consider n non-intersecting Brownian motions with two fixed starting positions and two fixed ending positions in the large n limit. We show that in case of 'large separation' between the endpoints, the particles are asymptotically distributed in two separate groups, with no interaction between them, as one would intuitively expect. We give a rigorous proof using the Riemann-Hilbert formalism. In the case of 'critical separation' between the endpoints we are led to a model Riemann-Hilbert problem associated to the Hastings-McLeod solution of the Painlevé II equation. We show that the Painlevé II equation also appears in the large n asymptotics of the recurrence coefficients of the multiple Hermite polynomials that are associated with the Riemann-Hilbert problem.

Multiple orthogonal polynomials (MOP) are a non-definite version of matrix orthogonal polynomials. They are described by a Riemann-Hilbert matrix Y consisting of four blocks In this paper, we show that det Y 1,1 (det Y 2,2 ) equals the average characteristic polynomial (average inverse characteristic polynomial, respectively) over the probabilistic ensemble that is associated to the MOP. In this way we generalize the classical results for orthogonal polynomials, and also some recent results for MOP of type I and type II. We then extend our results to arbitrary products and ratios of characteristic polynomials. In the latter case an important role is played by a matrix-valued version of the Christoffel-Darboux kernel. Our proofs use determinantal identities involving Schur complements, and adaptations of the classical results by Heine, Christoffel and Uvarov.