Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 103, 13 pages      arXiv:2504.06197      https://doi.org/10.3842/SIGMA.2025.103

Orthogonal Polynomials with Complex Densities and Quantum Minimal Surfaces

Giovanni Felder a and Jens Hoppe b
a) Department of Mathematics, ETH Zurich, 8092 Zurich, Switzerland
b) Technische Universität Braunschweig, Germany

Received September 01, 2025, in final form November 26, 2025; Published online December 07, 2025

Abstract
We show that the discrete Painlevé-type equations arising from quantum minimal surfaces are equations for recurrence coefficients of orthogonal polynomials for indefinite hermitian products. As a consequence, we obtain an explicit formula for the initial conditions leading to positive solutions.

Key words: orthogonal polynomials; quantum minimal surfaces; random matrices; Painlevé equations.

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