# 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} }

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…

## One Citation

Painlevé IV and the semi-classical Laguerre unitary ensembles with one jump discontinuities

- MathematicsAnalysis and Mathematical Physics
- 2021

In this paper, we present the characteristic of a certain discontinuous linear statistic of the semi-classical Laguerre unitary ensembles $$\begin{aligned} w(z,t)=A\theta (z-t)e^{-z^2+tz},…

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