On Airy solutions of P${}_{\mathrm{I}\mathrm{I}}$ and the complex cubic ensemble of random matrices, II

We describe the pole-free regions of the one-parameter family of special solutions of P${}_{\mathrm{I}\mathrm{I}}$, the second Painlevé equation, constructed from the Airy functions. This is achieved by exploiting the connection between these solutions and the recurrence coefficients of orthogonal polynomials that appear in the analysis of the ensemble of random matrices corresponding to the cubic potential.