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

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.