Fisher--Hartwig asymptotics and critical moments for the circular beta ensemble
Abstract
We determine the joint moments of real powers of the absolute value of characteristic polynomials in the circular beta ensemble, for every β>0. At fixed distinct prescribed points, the Fisher--Hartwig formula holds for every exponent greater than −1, with explicit constants and a relative error bounded by a negative power of the matrix size N. For approaching points in the same integrable range, the error is bounded by a negative power of NδN, provided that NδN→∞ and the sum of the absolute values of the negative exponents is less than two. Here δN is the minimum distance between the prescribed points; this range includes every integrable pair of root exponents. For q>0 at the critical relation pq2=2β, we prove the Keating--Wong asymptotic for the pth moment of the integral of the qth absolute power of the characteristic polynomial, for every real p>1. The leading asymptotic extends to every complex spatial weight in Lp. The proofs combine the recursion for orthogonal polynomials on the unit circle with conditional expectation identities for Gaussian multiplicative chaos and quantitative estimates for approaching singularities.
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BibTeX
@misc{chhaibi2026fisherhartwig,
title = {{Fisher--Hartwig asymptotics and critical moments for the circular beta ensemble}},
author = {Reda Chhaibi and Pierre-Loïc Méliot and Joseph Najnudel and Ashkan Nikeghbali},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.02205},
eprint = {2611.02205},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.02205},
}
RIS
TY - UNPB
TI - Fisher--Hartwig asymptotics and critical moments for the circular beta ensemble
AU - Reda Chhaibi
AU - Pierre-Loïc Méliot
AU - Joseph Najnudel
AU - Ashkan Nikeghbali
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.02205
UR - https://arxiv.org/abs/2611.02205
AB - We determine the joint moments of real powers of the absolute value of characteristic polynomials in the circular beta ensemble, for every β>0. At fixed distinct prescribed points, the Fisher--Hartwig formula holds for every exponent greater than −1, with explicit constants and a relative error bounded by a negative power of the matrix size N. For approaching points in the same integrable range, the error is bounded by a negative power of Nδ_N, provided that Nδ_N→∞ and the sum of the absolute values of the negative exponents is less than two. Here δ_N is the minimum distance between the prescribed points; this range includes every integrable pair of root exponents. For q>0 at the critical relation pq^2=2β, we prove the Keating--Wong asymptotic for the pth moment of the integral of the qth absolute power of the characteristic polynomial, for every real p>1. The leading asymptotic extends to every complex spatial weight in L^p. The proofs combine the recursion for orthogonal polynomials on the unit circle with conditional expectation identities for Gaussian multiplicative chaos and quantitative estimates for approaching singularities.
ER -