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