Skip to content
DraconiticResearch
Search Sources API
194,161 studies31 sourcesUpdated 7 min ago
PreprintarXiv licencearXiv

Asymptotics of Hankel determinants for potentials with singular edge points

  • Dan Dai
  • Jia-Hao Du
  • Chenhao Lu
Open at sourcePDFDOICite

Abstract

We establish the large-n asymptotics of Hankel determinants for unitary random matrix ensembles possessing a higher-order edge singularity. We focus on ensembles where the equilibrium measure is supported on a single interval and the limiting eigenvalue density vanishes to order 2k+1/2 for k ∈ ℕ. Notably, we explicitly evaluate the constant term in the asymptotic expansion, which involves a regularized integral of the Hamiltonian associated with the Painlevé I (PI2k) hierarchy. As a by-product, we also prove the universality of the eigenvalue correlation kernel near this singular edge and derive a limiting kernel expressed through functions related to a special solution of the PI2k equation. Our method relies on the Deift-Zhou nonlinear steepest descent analysis for the Riemann-Hilbert problem of orthogonal polynomials.

Related

  1. Asymptotics for likelihood ratio tests of boundary points with singular information and unidentifiable nuisance parameters
    Karl Oskar Ekvall, Ola Hössjer, Matteo Bottai +1
    arXivPreprintarXiv licenceFull textStatisticsarXiv
  2. Meromorphic order from Taylor coefficients, least singular values of Hankel matrices and rational approximants
    Kei Beauduin
    arXivPreprintCC BY-SA 4.0Full textAnalysis & PDEsarXiv
  3. Riordan Arrays and Shifted Hankel Determinants of OEIS A005773
    Daniel Yaqubi, Madjid Mirzavaziri
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  4. Hankel, Toeplitz, Hermitian--Toeplitz Determinants and the Zalcman Functional for a Class of Biholomorphic Mappings on Complex Banach Spaces
    Nabadwip Sarkar, Pradip Das
    arXivPreprintCC BY 4.0Full textAnalysis & PDEsarXiv
  5. Partial Results on Hankel Determinants and the Corresponding J-Fractions of a Sequence Related to Bernoulli Numbers
    Lin Jiu, Yihang Yin
    arXivPreprintarXiv licenceFull textAlgebra & number theoryCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{dai2026asymptotics,
  title = {{Asymptotics of Hankel determinants for potentials with singular edge points}},
  author = {Dan Dai and Jia-Hao Du and Chenhao Lu},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.26412},
  eprint = {2609.26412},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.26412},
}
RIS
TY  - UNPB
TI  - Asymptotics of Hankel determinants for potentials with singular edge points
AU  - Dan Dai
AU  - Jia-Hao Du
AU  - Chenhao Lu
PY  - 2026
DA  - 2026/09/22
PB  - arXiv
DO  - 10.48550/arxiv.2609.26412
UR  - https://arxiv.org/abs/2609.26412
AB  - We establish the large-n asymptotics of Hankel determinants for unitary random matrix ensembles possessing a higher-order edge singularity. We focus on ensembles where the equilibrium measure is supported on a single interval and the limiting eigenvalue density vanishes to order 2k+1/2 for k ∈ ℕ. Notably, we explicitly evaluate the constant term in the asymptotic expansion, which involves a regularized integral of the Hamiltonian associated with the Painlevé I (P_I^2k) hierarchy. As a by-product, we also prove the universality of the eigenvalue correlation kernel near this singular edge and derive a limiting kernel expressed through functions related to a special solution of the P_I^2k equation. Our method relies on the Deift-Zhou nonlinear steepest descent analysis for the Riemann-Hilbert problem of orthogonal polynomials.
ER  -