Painlevé representations for joint moments of characteristic polynomials from the orthogonal and unitary symplectic groups
Abstract
We study joint moments of characteristic polynomials and their second derivatives for Haar-distributed matrices from the unitary symplectic and orthogonal groups. We establish their large-N asymptotics throughout the full admissible range of the moment parameters and identify the limiting coefficients probabilistically in terms of the first inverse power sum of the Bessel point process. This extends the corresponding results of Assiotis, Gunes, Keating, and Wei [Proc. Lond. Math. Soc., 132 (2026), e70136] to the full admissible parameter range. Throughout this full range, we also express these limits in terms of a distinguished solution of the σ-Painlevé III' equation. Our approach reduces the problem to singular inverse-power statistics of the Jacobi unitary ensemble and combines their finite-N σ-Painlevé V representation with a refined Riemann-Hilbert analysis yielding precise small-x asymptotics and uniform derivative estimates for the finite-N correction term. A key ingredient in the probabilistic identification is a kernel-based proof that the rescaled first inverse power sum of the Jacobi ensemble converges in distribution to its Bessel point process counterpart.
Related
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- Integral Representations for Solutions of the Linearized Fourth Painleve Equation
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- Characteristic identities for orthogonal quantum matrices: examples
BibTeX
@misc{assiotis2026painlevé,
title = {{Painlevé representations for joint moments of characteristic polynomials from the orthogonal and unitary symplectic groups}},
author = {Theodoros Assiotis and Thomas Bothner and Jonathan P. Keating and Nicholas Simm and Fei Wei},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29947},
eprint = {2610.29947},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29947},
}
RIS
TY - UNPB
TI - Painlevé representations for joint moments of characteristic polynomials from the orthogonal and unitary symplectic groups
AU - Theodoros Assiotis
AU - Thomas Bothner
AU - Jonathan P. Keating
AU - Nicholas Simm
AU - Fei Wei
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29947
UR - https://arxiv.org/abs/2610.29947
AB - We study joint moments of characteristic polynomials and their second derivatives for Haar-distributed matrices from the unitary symplectic and orthogonal groups. We establish their large-N asymptotics throughout the full admissible range of the moment parameters and identify the limiting coefficients probabilistically in terms of the first inverse power sum of the Bessel point process. This extends the corresponding results of Assiotis, Gunes, Keating, and Wei [Proc. Lond. Math. Soc., 132 (2026), e70136] to the full admissible parameter range. Throughout this full range, we also express these limits in terms of a distinguished solution of the σ-Painlevé III' equation. Our approach reduces the problem to singular inverse-power statistics of the Jacobi unitary ensemble and combines their finite-N σ-Painlevé V representation with a refined Riemann-Hilbert analysis yielding precise small-x asymptotics and uniform derivative estimates for the finite-N correction term. A key ingredient in the probabilistic identification is a kernel-based proof that the rescaled first inverse power sum of the Jacobi ensemble converges in distribution to its Bessel point process counterpart.
ER -