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

Abstract

The Keating-Snaith central limit theorem proves that $\hat{I}{›}_N(A)=\log \det (I-A)$, for randomly drawn $A\in U(N)$, suitably normalised, tends to a complex Gaussian random variable in the large $N$ limit. The deviations of the real and imaginary parts of $\hat{I}{›}_N(A)$, 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 $\mathrm{Re}(\hat{I}{›}_N(A))$ and $\mathrm{Im}(\hat{I}{›}_N(A))$. Additionally our methods apply to the logarithm of the derivative of the characteristic polynomial evaluated at an eigenvalue of $A$.