Toeplitz determinants with a one-cut regular potential and Fisher–Hartwig singularities I. Equilibrium measure supported on the unit circle

Blackstone, Elliot;Charlier, Christophe;Lenells, Jonatan
(2024) Proceedings of the Royal Society of Edinburgh: Section A Mathematics — Vol. 154, n° 5, p. 1431-1472 (2024)

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(en) We consider Toeplitz determinants whose symbol has: (i) a one-cut regular potential V, (ii) Fisher–Hartwig singularities and (iii) a smooth function in the background. The potential V is associated with an equilibrium measure that is assumed to be supported on the whole unit circle. For constant potentials V, the equilibrium measure is the uniform measure on the unit circle and our formulas reduce to well-known results for Toeplitz determinants with Fisher–Hartwig singularities. For non-constant V, our results appear to be new even in the case of no Fisher–Hartwig singularities. As applications of our results, we derive various statistical properties of a determinantal point process which generalizes the circular unitary ensemble.
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Blackstone, E., Charlier, C., & Lenells, J. (2024). Toeplitz determinants with a one-cut regular potential and Fisher–Hartwig singularities I. Equilibrium measure supported on the unit circle. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 154(5), 1431-1472. https://doi.org/10.1017/prm.2023.73 (Original work published 2024)