# Unimodality for free multiplicative convolution with free normal distributions on the unit circle

## TL;DR

This paper investigates when free multiplicative convolution with free normal distributions on the unit circle results in unimodal distributions, providing conditions under which unimodality is preserved or lost, and comparing with classical convolution.

## Contribution

It establishes new conditions for unimodality preservation under free multiplicative convolution with free normal distributions on the unit circle.

## Key findings

- Symmetric unimodal distributions remain unimodal after convolution.
- Certain distributions become unimodal after large-time convolution.
- Bernoulli distribution remains non-unimodal at all times.

## Abstract

We study unimodality for free multiplicative convolution with free normal distributions $\{\lambda_t\}_{t>0}$ on the unit circle. We give four results on unimodality for $\mu\boxtimes\lambda_t$: (1) if $\mu$ is a symmetric unimodal distribution on the unit circle then so is $\mu\boxtimes \lambda_t$ at any time $t>0$; (2) if $\mu$ is a symmetric distribution on $\mathbb{T}$ supported on $\{e^{i\theta}: \theta \in [-\varphi,\varphi]\}$ for some $\varphi \in (0,\pi/2)$, then $\mu \boxtimes \lambda_t$ is unimodal for sufficiently large $t>0$; (3) ${\bf b} \boxtimes \lambda_t$ is not unimodal at any time $t>0$, where ${\bf b}$ is the equally weighted Bernoulli distribution on $\{1,-1\}$; (4) $\lambda_t$ is not freely strongly unimodal for sufficiently small $t>0$. Moreover, we study unimodality for classical multiplicative convolution (with Poisson kernels), which is useful in proving the above four results.