Benford's law and the C$\beta$E

TL;DR

This paper demonstrates that the leading digits of the characteristic polynomial of the Circular β-Ensemble follow Benford's Law as the matrix size grows large, with subsequent digits becoming uniformly distributed.

Contribution

It establishes the asymptotic digit distribution of the characteristic polynomial for the Circular β-Ensemble, including a convergence rate in the CLT for the logarithm of its absolute value.

Findings

Leading digits follow Benford's Law in the large N limit.

Further digits tend to be uniformly distributed.

Provides bounds on convergence rate in total variation norm.

Abstract

We study the individual digits for the absolute value of the characteristic polynomial for the Circular $\beta$-Ensemble. We show that, in the large $N$ limit, the first digits obey Benford's Law and the further digits become uniformly distributed. Key to the proofs is a bound on the rate of convergence in total variation norm in the CLT for the logarithm of the absolute value of the characteristic polynomial.