Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Painlev\'{e} IV, $\sigma-$Form and the Deformed Hermite Unitary
  Ensembles   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.

29
The categories of this paper are: funct-an
3
The categories of this paper are: physics
9
The categories of this paper are: nlin
23
The categories of this paper are: solv-int
24
The categories of this paper are: dg-ga
15
The categories of this paper are: nucl-ex
7
The categories of this paper are: hep-th
1
The categories of this paper are: cs
26
The categories of this paper are: bayes-an
11
The categories of this paper are: hep-lat
6
The categories of this paper are: astro-ph
35
The categories of this paper are: mtrl-th
17
The categories of this paper are: q-fin
36
The categories of this paper are: cmp-lg
37
The categories of this paper are: supr-con
12
The categories of this paper are: nucl-th