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

Abstract

The Keating-Snaith central limit theorem proves that $\Lambda_N(A)=\log\det(I-A)$, 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 $\Lambda_N(A)$, 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 $\operatorname{Re}(\Lambda_N(A))$ and $\operatorname{Im}(\Lambda_N(A))$. Additionally our methods apply to the logarithm of the derivative of the characteristic polynomial evaluated at an eigenvalue of $A$.