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

We study unimodality for free multiplicative convolution with free normal distributions $\{λ_t\}_{t>0}$ on the unit circle. We give four results on unimodality for $μ\boxtimesλ_t$: (1) if $μ$ is a symmetric unimodal distribution on the unit circle then so is $μ\boxtimes λ_t$ at any time $t>0$; (2) if $μ$ is a symmetric distribution on $\mathbb{T}$ supported on $\{e^{iθ}: θ\in [-φ,φ]\}$ for some $φ\in (0,π/2)$, then $μ\boxtimes λ_t$ is unimodal for sufficiently large $t>0$; (3) ${\bf b} \boxtimes λ_t$ is not unimodal at any time $t>0$, where ${\bf b}$ is the equally weighted Bernoulli distribution on $\{1,-1\}$; (4) $λ_t$ 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.