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arXiv · 2103.04229

Painlevé IV, $σ-$Form and the Deformed Hermite Unitary Ensembles

Mengkun Zhu·Dan Wang·Yang Chen
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Abstract

We study the Hankel determinant generated by a deformed Hermite weight with one jump $w(z,t,γ)=e^{-z^2+tz}|z-t|^γ(A+Bθ(z-t))$, where $A\geq 0$, $A+B\geq 0$, $t\in\textbf{R}$, $γ>-1$ and $z\in\textbf{R}$. 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 $α_n$, $β_n$, $R_n(t)$ and $r_n(t)$. Especially, we find that the auxiliary quantities $R_n(t)$ and $r_n(t)$ satisfy the coupled Riccati equations, and $R_n(t)$ satisfies a particular Painlevé IV equation. Based on above results, we show that $σ_n(t)$ and $\hatσ_n(t)$, two quantities related to the Hankel determinant and $R_n(t)$, satisfy the continuous and discrete $σ-$form equations, respectively. In the end, we also discuss the large $n$ asymptotic behavior of $R_n(t)$, 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.

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Mengkun Zhu, Dan Wang, Yang Chen. 2021-03-07. Painlevé IV, $σ-$Form and the Deformed Hermite Unitary Ensembles. https://doi.org/10.1063/5.0035471

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