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 $\mu ⊠{\lambda }_t$: (1) if $\mu$ is a symmetric unimodal distribution on the unit circle then so is $\mu ⊠{\lambda }_t$ at any time $t>0$; (2) if $\mu$ is a symmetric distribution on $\mathbb{T}$ supported on $\{e^{i\theta }:\theta \in [-\phi ,\phi ]\}$ for some $\phi \in (0,\pi /2)$, then $\mu ⊠{\lambda }_t$ is unimodal for sufficiently large $t>0$; (3) $\mathbf{b}⊠{\lambda }_t$ is not unimodal at any time $t>0$, where $\mathbf{b}$ 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.

arXiv comment19 pages, 4 figures