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.
Categories: math.PR