On the interim statistics for compact group characteristic polynomials and their derivatives

2025/10/01 by Erika Bailey, Bailey, E., Santiago Ortiz +1
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #FOS: Physical sciences #Geometry and complex manifolds #Mathematical Physics (math-ph) #Probability (math.PR) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2510.00675

openalex publication_date 2025/10/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The Keating-Snaith central limit theorem proves that Λ_N(A)=logdet(I-A), for randomly drawn A∈ U(N), suitably normalised, tends to a complex Gaussian random variable in the large N limit. The deviations of the real and imaginary parts of Λ_N(A), on the scale of a positive kth multiple of the variance, are known to be Gaussian but with a multiplicative perturbation in the form of the 2kth moment coefficient. Here we study the interpolating regime by allowing k=k(N) for both Re(Λ_N(A)) and Im(Λ_N(A)). Additionally our methods apply to the logarithm of the derivative of the characteristic polynomial evaluated at an eigenvalue of A.

Citations

• Complex moments of the derivative of the Riemann zeta function
• Lower Bounds for the Large Deviations of Selberg's Central Limit Theorem
• Large Deviation Estimates of Selberg's Central Limit Theorem and Applications
• Evidence of Random Matrix Corrections for the Large Deviations of Selberg's Central Limit Theorem
• Sharp upper bounds for fractional moments of the Riemann zeta function
• Sharp conditional bounds for moments of the Riemann zeta function
• A hybrid Euler-Hadamard product and moments of ζ'(ρ)
• Large deviations in Selberg's central limit theorem
• Moments of the Riemann zeta-function

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• A CLT for the characteristic polynomial of random Jacobi matrices, and the GβE