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

Blackstone, Elliot ; Charlier, Christophe LU and Lenells, Jonatan LU (2024) In Proceedings of the Royal Society of Edinburgh Section A: Mathematics 154(5). p.1431-1472
Abstract

We consider Toeplitz determinants whose symbol has: (i) a one-cut regular potential, (ii) Fisher-Hartwig singularities and (iii) a smooth function in the background. The potential is associated with an equilibrium measure that is assumed to be supported on the whole unit circle. For constant potentials, 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, 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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author
; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
asymptotics, Fisher-Hartwig singularities, Riemann-Hilbert problems, Toeplitz determinants
in
Proceedings of the Royal Society of Edinburgh Section A: Mathematics
volume
154
issue
5
pages
1431 - 1472
publisher
Royal Society of Edinburgh
external identifiers
  • scopus:85168996023
ISSN
0308-2105
DOI
10.1017/prm.2023.73
language
English
LU publication?
yes
id
85be254b-5510-4e14-8cbf-11fbf10af6c6
date added to LUP
2023-11-10 13:22:27
date last changed
2024-12-17 16:05:20
@article{85be254b-5510-4e14-8cbf-11fbf10af6c6,
  abstract     = {{<p>We consider Toeplitz determinants whose symbol has: (i) a one-cut regular potential, (ii) Fisher-Hartwig singularities and (iii) a smooth function in the background. The potential is associated with an equilibrium measure that is assumed to be supported on the whole unit circle. For constant potentials, 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, 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.</p>}},
  author       = {{Blackstone, Elliot and Charlier, Christophe and Lenells, Jonatan}},
  issn         = {{0308-2105}},
  keywords     = {{asymptotics; Fisher-Hartwig singularities; Riemann-Hilbert problems; Toeplitz determinants}},
  language     = {{eng}},
  number       = {{5}},
  pages        = {{1431--1472}},
  publisher    = {{Royal Society of Edinburgh}},
  series       = {{Proceedings of the Royal Society of Edinburgh Section A: Mathematics}},
  title        = {{Toeplitz determinants with a one-cut regular potential and Fisher-Hartwig singularities I. Equilibrium measure supported on the unit circle}},
  url          = {{http://dx.doi.org/10.1017/prm.2023.73}},
  doi          = {{10.1017/prm.2023.73}},
  volume       = {{154}},
  year         = {{2024}},
}