Painlevé IV, $σ-$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, γ) = e −z 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 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 σn (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.