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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.