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

Abstract

We study the Hankel determinant generated by a deformed Hermite weight with one jump $w(z,t,\gamma )=e^{-z^2+tz}|z-t{|}^{\gamma }(A+B\theta (z-t))$, where $A\ge 0$, $A+B\ge 0$, $t\in \text{R}$, $\gamma >-1$ and $z\in \text{R}$. 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 ${\alpha }_n$, ${\beta }_n$, $R_n(t)$ 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 ${\sigma }_n(t)$ and ${\hat{\sigma }}_n(t)$, two quantities related to the Hankel determinant and $R_n(t)$, satisfy the continuous and discrete $\sigma -$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.

References in corpus (2)

• Painleve IV and degenerate Gaussian Unitary Ensembles
• Orthogonal polynomials with discontinuous weights