• Corpus ID: 232147041

Painlev\'{e} IV, $\sigma-$Form and the Deformed Hermite Unitary Ensembles

@inproceedings{Zhu2021PainleveI,
  title={Painlev\'\{e\} IV, \$\sigma-\$Form and the Deformed Hermite Unitary Ensembles},
  author={Mengkun Zhu and Dan Wang and Yang Chen},
  year={2021}
}

• Mengkun Zhu, Dan Wang, Yang Chen
• Published 7 March 2021
• Mathematics

We study the Hankel determinant generated by a deformed Hermite weight with one jump w(z, t, γ) = e 2+tz|z − t|γ(A + Bθ(z − t)), where A ≥ 0, A + B ≥ 0, t ∈ R, γ > −1 and z ∈ R. By using the ladder operators for the corresponding monic orthogonal polynomials, and their relative compatibility conditions, we obtain a series of difference and differential equations to describe the relations among αn, βn, Rn(t) and rn(t). Especially, we find that the auxiliary quantities Rn(t) and rn(t) satisfy the coupled Riccati equations, and Rn(t) satisfies a particular Painlevé IV equation. Based on above results, we show that σn(t) and σ̂n(t), two quantities related to the Hankel determinant and Rn(t), satisfy the continuous and discrete σ−form equations, respectively. In the end, we also discuss the large n asymptotic behavior of Rn(t), which produce the expansion of the logarithmic of the Hankel determinant and the asymptotic of the second order differential equation of the monic orthogonal polynomials.