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

E. Bailey, S. Ortiz

TL;DR

The paper advances the understanding of interim statistics for compact-group characteristic polynomials by introducing an interpolating regime between the Keating–Snaith CLT and large-deviation behavior. It shows how letting the deviation index $k$ grow with the matrix size, via a parameter $\alpha$, yields density formulas for $\Re\log P_N(A,\theta)$ and $\Im\log P_N(A,\theta)$ that either pick up multiplicative moment coefficients $c_\kappa$ (for $\alpha>1$) or reduce to Gaussian form (for $\alpha\in[0,1)$), with a smoothing to resolve a sharp transition. The results extend to the derivative $P_N'(A,\theta_1)$, giving analogous interpolating densities with constants $f_\kappa$, $g_\kappa$ and establishing CLTs and independence in the large-$N$ limit; these findings connect to number-theoretic analogies for the Riemann zeta function and its derivative. Overall, the work provides a unified, quantitatively precise framework for tail behavior of both the characteristic polynomial and its derivative across regimes, with potential implications for zeta-model predictions and related random-matrix models.

Abstract

The Keating-Snaith central limit theorem proves that $Λ_N(A)=\log\det(I-A)$, for randomly drawn $A\in \operatorname{U}(N)$, 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 $k$th multiple of the variance, are known to be Gaussian but with a multiplicative perturbation in the form of the $2k$th moment coefficient. Here we study the interpolating regime by allowing $k=k(N)$ for both $\operatorname{Re}(Λ_N(A))$ and $\operatorname{Im}(Λ_N(A))$. Additionally our methods apply to the logarithm of the derivative of the characteristic polynomial evaluated at an eigenvalue of $A$.

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

TL;DR

The paper advances the understanding of interim statistics for compact-group characteristic polynomials by introducing an interpolating regime between the Keating–Snaith CLT and large-deviation behavior. It shows how letting the deviation index grow with the matrix size, via a parameter , yields density formulas for and that either pick up multiplicative moment coefficients (for ) or reduce to Gaussian form (for ), with a smoothing to resolve a sharp transition. The results extend to the derivative , giving analogous interpolating densities with constants , and establishing CLTs and independence in the large- limit; these findings connect to number-theoretic analogies for the Riemann zeta function and its derivative. Overall, the work provides a unified, quantitatively precise framework for tail behavior of both the characteristic polynomial and its derivative across regimes, with potential implications for zeta-model predictions and related random-matrix models.

Abstract

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

Paper Structure

This paper contains 8 sections, 7 theorems, 85 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Results
  4. Proofs
  5. Proof of Theorem \ref{['thm1']}
  6. Proof of Theorem \ref{['thm2']}
  7. Proof of Theorem \ref{['thm3']}
  8. Proof of Theorems \ref{['thm4']} and \ref{['thm5']}

Key Result

Theorem 1.1

Let $\theta\in\mathbb{R}$ and draw $A\in \mathop{\mathrm{U}}\nolimits(N)$ with respect to Haar measure. Write $\rho_N$ for the probability density function for $\mathop{\mathrm{Re}}\nolimits\log P_N(A,\theta)/\sqrt{Q_2(N)}$ where is the second cumulant of $\mathop{\mathrm{Re}}\nolimits\log P_N(A,\theta)$. Set, for $\alpha\geq0$, and $\kappa>0$ fixed Then as $N\rightarrow\infty$, writing $n=\log\

Figures (1)

  • Figure 1: Histograms of (a) $\log|P_N(A,0)|$ and (b) $\log|P_N'(A,\theta_1)|$, both scaled, for $N=75$, with $5,000$ data points, against the standard Normal distribution (solid line).

Theorems & Definitions (12)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark
  • Remark
  • Remark
  • Lemma 2.1
  • proof
  • ...and 2 more