# Random characters under the $L$-measure, I : Dirichlet characters

## TL;DR

This paper introduces an $L$-measure on Dirichlet characters, compares it with the Plancherel measure, and studies the asymptotic distribution of character evaluations, revealing convergence to imaginary exponentials of a Cauchy distribution.

## Contribution

It defines a new $L$-measure on Dirichlet characters and analyzes the limiting distribution of character evaluations as the group size increases.

## Key findings

- Character evaluations converge to imaginary exponentials of a Cauchy distribution.
- Contrast with symmetric group characters, which converge to Gaussians.
- The $L$-measure provides a new perspective on the distribution of Dirichlet characters.

## Abstract

We define the $L$-measure on the set of Dirichlet characters as an analogue of the Plancherel measure, once considered as a measure on the irreducible characters of the symmetric group. We compare the two measures and study the limit in distribution of characters evaluations when the size of the underlying group grows. These evaluations are proven to converge in law to imaginary exponentials of a Cauchy distribution in the same way as the rescaled windings of the complex Brownian motion. This contrasts with the case of the symmetric group where the renormalised characters converge in law to Gaussians after rescaling (Kerov Central Limit Theorem).