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Toeplitz determinants with a one-cut regular potential and Fisher–Hartwig singularities I. Equilibrium measure supported on the unit circle
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 154 / Issue 5 / October 2024
- Published online by Cambridge University Press:
- 15 August 2023, pp. 1431-1472
- Print publication:
- October 2024
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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.
