Universality and critical behaviour in the chiral two-matrix model

Abstract

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) = y2/2 + αy 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 × 4 matrix-valued Riemann–Hilbert problem associated with the Painlevé II equation q″(x) = xq (x) + 2q3(x) − ν − 1/2. In this way we obtain the chiral analogue of a recent result by Duits and the second author.