Spectral extremes under exact cycle conditioning
Abstract
Let Pn be the matrix of a random permutation of n symbols and let Mn=logmax|z|=1|det(I−zPn)|. Cook and Zeitouni proved that Mn/log n converges in probability to a constant x0 for a uniform permutation. We show that the √(log n) fluctuations of Mn are carried entirely by the number of cycles Kn. Write λ(s)=log{Γ(1+s)/Γ(1+s/2)2}, let sκ minimize (1+κλ(s))/s on (0,∞), and put v(κ)=κλ'(sκ) and aθ=λ(sθ)/sθ. Under the Ewens measure with any fixed parameter θ>0 we prove Mn=v(θ)log n+aθ(Kn−θlog n)+OP(loglog n), so that the standardized pair (Kn,Mn) converges jointly to (G,G) with G standard normal: the maximum and the cycle count are asymptotically perfectly aligned. This is deduced from a statement about the exact conditional law, which does not depend on θ: for every compact [κ-,κ+]⊂(0,∞) there is a finite C such that P(|Mn−v(k/log n)log n|>Cloglog n | Kn=k) tends to $0$ uniformly over integers k with κ-log n≤ k≤κ+log n, that is, over exact and possibly atypical cycle counts. The proof keeps the size and the cycle count simultaneously in a two-variable coefficient extraction. Cycles longer than n/(log n)4 are reserved as an analytic factor whose coefficients are flat under every size shift produced by the shorter cycles; positivity then converts a scalar coefficient asymptotic into a relative comparison of the entire path-constrained measure, with an error that does not degrade with the number of constraints or with the rarity of the event. The constrained lower bound comes from pointwise saddle estimates for killed convolutions along a dyadic chain of endpoint boxes.
Related
- Spectral Measure and Tail Process for Multivariate Dynamical Extremes
- Extremes of solar spectral irradiance in the SORCE/XPS record
- Exploiting Exact Conditionals Improves Conditioning: Provably Fast Mixing Time Bounds By Sampling from the Marginal
- An A-B-C Coordinate Encoding for Exact Boundary-Cycle Extraction on Hexagonal Lattices
- On the Spectral Region of 4-Cycle Stochastic Matrices
BibTeX
@misc{lu2026spectral,
title = {{Spectral extremes under exact cycle conditioning}},
author = {Zhipeng Lu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13331},
eprint = {2609.13331},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13331},
}
RIS
TY - UNPB
TI - Spectral extremes under exact cycle conditioning
AU - Zhipeng Lu
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13331
UR - https://arxiv.org/abs/2609.13331
AB - Let P_n be the matrix of a random permutation of n symbols and let M_n=logmax_|z|=1|det(I−zP_n)|. Cook and Zeitouni proved that M_n/log n converges in probability to a constant x_0 for a uniform permutation. We show that the √(log n) fluctuations of M_n are carried entirely by the number of cycles K_n. Write λ(s)=log{Γ(1+s)/Γ(1+s/2)^2}, let s_κ minimize (1+κλ(s))/s on (0,∞), and put v(κ)=κλ'(s_κ) and a_θ=λ(s_θ)/s_θ. Under the Ewens measure with any fixed parameter θ>0 we prove M_n=v(θ)log n+a_θ(K_n−θlog n)+O_P(loglog n), so that the standardized pair (K_n,M_n) converges jointly to (G,G) with G standard normal: the maximum and the cycle count are asymptotically perfectly aligned. This is deduced from a statement about the exact conditional law, which does not depend on θ: for every compact [κ_-,κ_+]⊂(0,∞) there is a finite C such that P(|M_n−v(k/log n)log n|>Cloglog n | K_n=k) tends to $0$ uniformly over integers k with κ_-log n≤ k≤κ_+log n, that is, over exact and possibly atypical cycle counts. The proof keeps the size and the cycle count simultaneously in a two-variable coefficient extraction. Cycles longer than n/(log n)^4 are reserved as an analytic factor whose coefficients are flat under every size shift produced by the shorter cycles; positivity then converts a scalar coefficient asymptotic into a relative comparison of the entire path-constrained measure, with an error that does not degrade with the number of constraints or with the rarity of the event. The constrained lower bound comes from pointwise saddle estimates for killed convolutions along a dyadic chain of endpoint boxes.
ER -