Large-parameter asymptotics of generalized Hastings–McLeod functions

The generalized Hastings–McLeod solutions to the inhomogeneous Painlevé-II equation arise in multi-critical unitary random matrix ensembles, the chiral two-matrix model for rectangular matrices, non-intersecting squared Bessel paths, and non-intersecting Brownian bridges on the circle. We establish the leading-order asymptotic behavior of the generalized Hastings–McLeod functions as the inhomogeneous parameter approaches infinity using the Deift–Zhou nonlinear steepest-descent method for Riemann–Hilbert problems. This analysis is done in both the pole-free region and pole region. The asymptotic formulae show excellent agreement with numerically computed solutions in both regions.