# Nedialko Bradinoff: Benford’s law and the circular beta ensemble

**Time: **
Tue 2023-03-21 15.15 - 16.15

**Location: **
KTH, room 3721

**Participating: **
Nedialko Bradinoff (KTH)

**Abstract.**

Benford's law is a remarkable phenomenon that governs the digital expansion of both deterministic and random real quantities, coming from a broad variety of contexts, including terms of geometric progressions, stock prises, population numbers… Roughly, the leading digits of such quantities are likely to be heavily skewed towards smaller values. The focus of this talk is to describe a connection between Benford's law and another beautiful object, the Circular $\beta$-Ensemble. The C$\beta$E(N) is classical in random matrix theory and has been studied extensively to exhibit universal properties. A natural object that is studied in the context of the C$\beta$E(N) is its characteristic polynomial. In this talk I will describe how its absolute value obeys Benford's law in a strong sense as $N$ tends to infinity. Along the way I will discuss sufficient conditions for such a result and illustrate their necessity with some examples. Key to the discussed result is a central limit theorem in total variation norm of the logarithm of the absolute value of the characteristic polynomial. The talk is based on joint work with Maurice Duits.