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

TL;DR

This paper investigates the properties of Hankel determinants generated by a deformed Hermite weight with a jump, deriving differential and difference equations, including a Painlevé IV equation, and analyzing their asymptotic behavior.

Contribution

It introduces a novel analysis of Hankel determinants with a jump deformation, deriving new coupled Riccati and Painlevé IV equations, and explores their asymptotics.

Findings

Riccati equations for auxiliary quantities R_n(t) and r_n(t)

R_n(t) satisfies a Painlevé IV equation

Asymptotic expansion of the Hankel determinant logarithm

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 $A\geq 0$, $A+B\geq 0$, $t\in\textbf{R}$, $\gamma>-1$ 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 $\alpha_n$, $\beta_n$, $R_n(t)$ 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 Painlev\'{e} 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.