The double scaling limit of an ensemble of random unitary \(n \times n\) matrices near singular points where a new cut is emerging from the support of the equilibrium measure is studied. Using the ansatz obtained by B. Eynard [J. Stat. Mech. 7, P07005 (2006)]) an approximated equilibrium density is constructed satisfying some conditions outside neighborhoods of the edge points and the critical point.
Error terms in these conditions are analyzed. The Deift-Zhou steepest descent method to the Riemann-Hilbert problem of orthogonal polynomials is applied. The approximated density is used to construct a “\(g\)-function” for modification of the Riemann-Hilbert problem. The modified Riemann-Hilbert problem provides parameters of the asymptotic of the orthogonal polynomials.
Finally, it is proved that asymptotic behavior of the kernel near the central point is given by the correlation kernel of a random unitary matrix ensemble with weight \(e^{-x^{2\nu}}\), where \(\nu\) is positive integer.