The Riemann-Hilbert approach to obtain critical asymptotics for Hamiltonian perturbations of hyperbolic and elliptic systems

The present paper gives an overview of the recent developments in the description of critical behavior for Hamiltonian perturbations of hyperbolic and elliptic systems of partial differential equations. It was conjectured that this behavior can be described in terms of distinguished Painlev\'e transcendents, which are universal in the sense that they are, to some extent, independent of the equation and the initial data. We will consider several examples of well-known integrable equations that are expected to show this type of Painlev\'e behavior near critical points. The Riemann-Hilbert method is a useful tool to obtain rigorous results for such equations. We will explain the main lines of this method and we will discuss the universality conjecture from a Riemann-Hilbert point of view.Comment: review paper, 22 pages, 4 figure

Random matrices with equispaced external source

We study Hermitian random matrix models with an external source matrix which has equispaced eigenvalues, and with an external field such that the limiting mean density of eigenvalues is supported on a single interval as the dimension tends to infinity. We obtain strong asymptotics for the multiple orthogonal polynomials associated to these models, and as a consequence for the average characteristic polynomials. One feature of the multiple orthogonal polynomials analyzed in this paper is that the number of orthogonality weights of the polynomials grows with the degree. Nevertheless we are able to characterize them in terms of a pair of 2 x 1 vector-valued Riemann-Hilbert problems, and to perform an asymptotic analysis of the Riemann-Hilbert problems.Comment: 53 pages, 9 figures; textual changes and minor typos correcte

Biorthogonal ensembles with two-particle interactions

We investigate determinantal point processes on $[0,+\infty)$ of the form \begin{equation*}\label{probability distribution} \frac{1}{Z_n}\prod_{1\leq i<j\leq n}(\lambda_j-\lambda_i)\prod_{1\leq i<j\leq n}(\lambda_j^\theta-\lambda_i^\theta) \prod_{j=1}^n w(\lambda_j)d\lambda_j,\qquad \theta\geq 1. \end{equation*} We prove that the biorthogonal polynomials associated to such models satisfy a recurrence relation and a Christoffel-Darboux formula if $\theta\in\mathbb Q$, and that they can be characterized in terms of $1\times 2$ vector-valued Riemann-Hilbert problems which exhibit some non-standard properties. In addition, we obtain expressions for the equilibrium measure associated to our model if $w(\lambda)=e^{-nV(\lambda)}$ in the one-cut case with and without hard edge.Comment: 28 pages, 6 figure

Asymptotics for Toeplitz determinants: perturbation of symbols with a gap

We study the determinants of Toeplitz matrices as the size of the matrices tends to infinity, in the particular case where the symbol has two jump discontinuities and tends to zero on an arc of the unit circle at a sufficiently fast rate. We generalize an asymptotic expansion by Widom [22], which was known for symbols supported on an arc. We highlight applications of our results in the Circular Unitary Ensemble and in the study of Fredholm determinants associated to the sine kernel.Comment: 28 pages, 5 figure

Critical asymptotic behavior for the Korteweg-de Vries equation and in random matrix theory

We discuss universality in random matrix theory and in the study of Hamiltonian partial differential equations. We focus on universality of critical behavior and we compare results in unitary random matrix ensembles with their counterparts for the Korteweg-de Vries equation, emphasizing the similarities between both subjects.Comment: review paper, 19 pages, to appear in the proceedings of the MSRI semester on `Random matrices, interacting particle systems and integrable systems

Random Matrices with Merging Singularities and the Painlev\'e V Equation

We study the asymptotic behavior of the partition function and the correlation kernel in random matrix ensembles of the form $\frac{1}{Z_n} \big|\det \big( M^2-tI \big)\big|^{\alpha} e^{-n\operatorname{Tr} V(M)}dM$, where $M$ is an $n\times n$ Hermitian matrix, $\alpha>-1/2$ and $t\in\mathbb R$, in double scaling limits where $n\to\infty$ and simultaneously $t\to 0$. If $t$ is proportional to $1/n^2$, a transition takes place which can be described in terms of a family of solutions to the Painlev\'e V equation. These Painlev\'e solutions are in general transcendental functions, but for certain values of $\alpha$, they are algebraic, which leads to explicit asymptotics of the partition function and the correlation kernel

Hankel determinant and orthogonal polynomials for a Gaussian weight with a discontinuity at the edge

We compute asymptotics for Hankel determinants and orthogonal polynomials with respect to a discontinuous Gaussian weight, in a critical regime where the discontinuity is close to the edge of the associated equilibrium measure support. Their behavior is described in terms of the Ablowitz-Segur family of solutions to the Painlev\'e II equation. Our results complement the ones in [Xu,Zhao,2011]. As consequences of our results, we conjecture asymptotics for an Airy kernel Fredholm determinant and total integral identities for Painlev\'e II transcendents, and we also prove a new result on the poles of the Ablowitz-Segur solutions to the Painlev\'e II equation. We also highlight applications of our results in random matrix theory.Comment: 35 pages, 4 figure