Universality and critical behavior 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 $W(y)=y^2/2+\alpha y$ 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 Painlevé II equation $q"(x)=xq(x)+2q^3(x)-\nu -1/2$. In this way we obtain the chiral analogue of a recent result by Duits and the second author.

70 pages, 10 figures

References in corpus (6)

• A new Chiral Two-Matrix Theory for Dirac Spectra with Imaginary Chemical Potential
• Weakly Admissible Vector Equilibrium Problems
• Parameters of the lowest order chiral Lagrangian from fermion eigenvalues
• Microscopic eigenvalue correlations in QCD with imaginary isospin chemical potential
• A vector equilibrium problem for the two-matrix model in the quartic/quadratic case
• Non-intersecting squared Bessel paths: critical time and double scaling limit

Cited by in corpus (1)

• Singular values for products of two coupled random matrices: hard edge phase transition