Universality and critical behavior in the chiral two-matrix model

We study the chiral two-matrix model with polynomial potential functions $V$ and $W$, which was introduced by Akemann, Damgaard, Osborn and Splittorff. We show that the squared singular values of each of the individual matrices in this model form a determinantal point process with correlation kernel determined by a matrix-valued Riemann-Hilbert problem. The size of the Riemann-Hilbert matrix depends on the degree of the potential function $W$ (or $V$ respectively). In this way we obtain the chiral analogue of a result of Kuijlaars-McLaughlin for the non-chiral two-matrix model. The Gaussian case corresponds to $V,W$ being linear. For the case where $W(y)=y^2\mathrm{/}2+\alpha y$ is quadratic, we derive the large $n$-asymptotics of the Riemann-Hilbert problem by means of the Deift-Zhou steepest descent method. This proves universality in this case. An important ingredient in the analysis is a third-order differential equation. Finally we show that if also $V(x)=x$ is linear, then a multi-critical limit of the kernel exists which is described by a $4\times 4$ matrix-valued Riemann-Hilbert problem associated to the Painlev'e II equation $q\mathrm{"}(x)=xq(x)+2q^3(x)-\nu -1\mathrm{/}2$. In this way we obtain the chiral analogue of a recent result by Duits and the second author.