• Corpus ID: 232147041

# 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}
}
• Published 7 March 2021
• Mathematics
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…
1 Citations
Painlevé IV and the semi-classical Laguerre unitary ensembles with one jump discontinuities
• Mathematics
Analysis 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},

## References

SHOWING 1-10 OF 29 REFERENCES
Painlevé transcendents and the Hankel determinants generated by a discontinuous Gaussian weight
• Mathematics
Mathematical Methods in the Applied Sciences
• 2018
This paper studies the Hankel determinants generated by a discontinuous Gaussian weight with one and two jumps. It is an extension in a previous study, in which they studied the discontinuous
Fredholm determinants
The article provides with a down to earth exposition of the Fredholm theory with applications to Brownian motion and KdV equation.
On Semi-classical Orthogonal Polynomials Associated with a Freud-type Weight
• Math. Meth. Appl. Sci., 43
• 2020
Orthogonal polynomials, bi-confluent Heun equations and semi-classical weights
• Mathematics
• 2020
In this paper, we focus on four weights where , , , N>0; where , , ; with , , , , where is the Heaviside step function; and with , , N>0, . The second-order differential equations satisfied by , the
Orthogonal polynomials, asymptotics, and Heun equations
• Mathematics
Journal of Mathematical Physics
• 2019
The Painleve equations arise from the study of Hankel determinants generated by moment matrices, whose weights are expressed as the product of classical" weights multiplied by suitable
Center of mass distribution of the Jacobi unitary ensembles: Painlevé V, asymptotic expansions
• Mathematics
Journal of Mathematical Physics
• 2018
In this paper, we study the probability density function, $\mathbb{P}(c,\alpha,\beta, n)\,dc$, of the center of mass of the finite $n$ Jacobi unitary ensembles with parameters $\alpha\,>-1$ and
Center ofmass distribution of the Jacobi unitary ensembles: Painlevé V
• asymptotic expansions, J. Math. Phys., 59(10)
• 2018
On properties of a deformed Freud weight
• Mathematics
Random Matrices: Theory and Applications
• 2018
We study the recurrence coefficients of the monic polynomials [Formula: see text] orthogonal with respect to the deformed (also called semi-classical) Freud weight [Formula: see text] with parameters
Random matrix models
• doubletime Painlevé equations, and wireless relaying, J. Phys. A.: Math. Gen., 54
• 2013