We study unimodality for free multiplicative convolution with free normal
distributions {λt​}t>0​ on the unit circle. We give four results on
unimodality for μ⊠λt​: (1) if μ is a symmetric unimodal
distribution on the unit circle then so is μ⊠λt​ at any time
t>0; (2) if μ is a symmetric distribution on T supported on
{eiθ:θ∈[−φ,φ]} for some φ∈(0,π/2), then μ⊠λt​ is unimodal for sufficiently large
t>0; (3) b⊠λt​ is not unimodal at any time t>0,
where b is the equally weighted Bernoulli distribution on {1,−1};
(4) λt​ 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.Comment: 19 pages, 4 figure