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4. Random Matrix Theory

    1.     Please see Alfredo Deaño and Nick Simm’s article "Characteristic polynomials of complex random matrices and Painlevè transcendents" for background on this problem.

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      Problem 4.1.

      [Tamara Grava] Study the expansion of partition functions of normal matrix models for \[v(z,\overline{z})=(z \overline{z})^d-t(z^d+\overline{z}^d)\] and \[z_N=C(d,N)\prod_{\mathscr{l}=0}^{d-1} \mathbb{E}_{\text{Gin}}\Big\{\Big|\text{det}(M_{\text{Gin}}-z)\Big|^{\gamma_{\mathscr{l}}}\Big\},\] where \(z=t\sqrt{d}\). Study the "mad criticality" that happens here.


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          Problem 4.2.

          [Maurice Duits] Study the topological phase transition in the asymptotics of planar orthogonal polynomials with normal Random Matrix Theory.

              Cite this as: AimPL: Riemann-Hilbert problems, Toeplitz matrices, and applications, available at http://aimpl.org/riemhilberttoeplitz.