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PreprintCC BY 4.0arXiv

Approximating the poles of a Painlevé V transcendent for the Fredholm determinant of the sine kernel

  • Walter Van Assche
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Abstract

The Fredholm determinant of the sine kernel has an expression as an integral containing a Painlevé transcendent r satisfying Painlevé V, in the same way that the Fredholm determinant of the Airy kernel has an expression involving the Hastings-McLeod solution of Painlevé II. In this paper we use Padé approximation to approximate the poles of this Painlevé transcendent r in the complex plane. We use Padé approximation near $0$, Hermite-Padé approximation near $0$ and two-point Padé approximation near $0$ and infinity.

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Cite

BibTeXRIS
BibTeX
@misc{assche2026approximating,
  title = {{Approximating the poles of a Painlevé V transcendent for the Fredholm determinant of the sine kernel}},
  author = {Walter Van Assche},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38625},
  eprint = {2610.38625},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38625},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Approximating the poles of a Painlevé V transcendent for the Fredholm determinant of the sine kernel
AU  - Walter Van Assche
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38625
UR  - https://arxiv.org/abs/2610.38625
AB  - The Fredholm determinant of the sine kernel has an expression as an integral containing a Painlevé transcendent r satisfying Painlevé V, in the same way that the Fredholm determinant of the Airy kernel has an expression involving the Hastings-McLeod solution of Painlevé II. In this paper we use Padé approximation to approximate the poles of this Painlevé transcendent r in the complex plane. We use Padé approximation near $0$, Hermite-Padé approximation near $0$ and two-point Padé approximation near $0$ and infinity.
ER  -