$\newcommand{\Cat}{{\rm Cat}}$ $\newcommand{\A}{\mathcal A}$ $\newcommand{\freestar}{ \framebox[7pt]{\star} }$

## 2. Expectations of Characteristic Polynomials

1. #### Problem 2.1.

Identify whether or not there exists an integrable system (by the usual accepted criteria) underlying $\mathbb{E}_{A\in U(N)}\left(|Z|^{2k-2h}|Z^\prime|^{2h}\right),$

where $Z=\det(I-\lambda A)$.
• #### Problem 2.2.

Consider the expectation given in 2.1. For $h=k$, formulate a conjecture for $k\in\mathbb{R}$
• #### Problem 2.3.

For $h=k$, is there a Painlevé equation related to the second term?
• #### Problem 2.4.

Now for $h=0$, is there a connection with Painlevé?
• #### Problem 2.5.

How would the conclusions change if one replaces $\mathbb{E}_{A\in U(N)}$ by an average over the other classical groups?
• #### Problem 2.6.

Calculate $\mathbb{E}\left|\frac{Z\prime}{Z}\right|^{2k}_{\theta+i\varepsilon}$ - is this related to Painlevé?

Cite this as: AimPL: Painleve equations and their applications, available at http://aimpl.org/painleveapp.