Universality and critical behaviour in the chiral two-matrix model (Q2845636)

This paper is motivated by the determination of the pion decay constant with the help of lattice simulations and random maxtrix theory. The authors study a specific model, the so-called quenched chiral two-matrix model. To understand this model, one can start from the simplest version NEWLINE\[NEWLINE \frac{1}{\hat{Z}_n} \exp\left(-n \text{Tr}\left(\Phi^\ast \Phi + \Psi^\ast \Psi \right) \right) d\Phi d\Psi, NEWLINE\]NEWLINE where \(\Phi\) and \(\Psi\) are rectangular complex matrices of size \(n\times(n+v)\), \(\hat{Z}_n\) is a normalisation constant and \(d\Phi\) and \(d\Psi\) are the flat complex Lebesgue measures on the entries of \(\Phi\) and \(\Psi\). With the change of variables NEWLINE\[NEWLINE \Phi_1 = \Phi + \mu_1 \Psi, \;\;\; \Phi_2 = \Phi + \mu_2 \Psi, \;\;\; \mu_1, \mu_2 \in {\mathbb R} NEWLINE\]NEWLINE one arrives at NEWLINE\[NEWLINE \frac{1}{Z_n} \exp\left(-n \text{Tr}\left(c_1 \Phi_1^\ast \Phi_1 + c_2 \Phi_2^\ast \Phi_2 - \tau \left( \Psi_1^\ast \Psi_2 + \Psi_2^\ast \Psi_1 \right) \right) \right) d\Phi d\Psi, NEWLINE\]NEWLINE where \(c_1\), \(c_2\) and \(\tau\) are known functions of \(\mu_1\) and \(\mu_2\). One requires \(c_1,c_2 > 0\) and \(0 < \tau^2 < c_1 c_2\). The model considered by the authors is now a generalisation of this. They consider NEWLINE\[NEWLINE \frac{1}{Z_n} \exp\left(-n \text{Tr}\left( V\left(\Phi_1^\ast \Phi_1\right) + W\left(\Phi_2^\ast \Phi_2\right) - \tau \left( \Psi_1^\ast \Psi_2 + \Psi_2^\ast \Psi_1 \right) \right) \right) d\Phi d\Psi, NEWLINE\]NEWLINE where \(V\) and \(W\) are polynomials with positive leading coefficients.NEWLINENEWLINEIf both polynomials are linear \(V(x)=c_1 x\), \(W(y)=c_2 y\) one is back to the previous (Gaussian) case. In this paper the authors discuss the case NEWLINE\[NEWLINE W(y) = \frac{1}{2} y^2 + \alpha y, \;\;\; \alpha \in {\mathbb R}NEWLINE\]NEWLINE and \(V(x)\) arbitrary. The authors relate the correlation kernel to a Riemann-Hilbert problem and prove for the above mentioned choice of \(W(y)\) universality. They then specialise to a specific choice of \(V(x)\). Taking the polynomial \(V(x)\) as linear \(V(x)=x\), they discuss in detail the \(\alpha\tau\)-phase diagram and phase transitions.