On Airy Solutions of P_II and Complex Cubic Ensemble of Random Matrices, I

2023/10/23 by Barhoumi, Ahmad, Bleher, Pavel, Deaño, Alfredo +1
#15B52 #33C10 #33C47 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2310.14898

Abstract

We show that the one-parameter family of special solutions of P_II, the second Painlevé equation, constructed from the Airy functions, as well as associated solutions of P_XXXIV and S_II, can be expressed via the recurrence coefficients of orthogonal polynomials that appear in the analysis of the Hermitian random matrix ensemble with a cubic potential. Exploiting this connection we show that solutions of P_II that depend only on the first Airy function Ai (but not on Bi ) possess a scaling limit in the pole free region, which includes a disk around the origin whose radius grows with the parameter. We then use the scaling limit to show that these solutions are monotone in the parameter on the negative real axis.

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