On the Free Convolution with a Free Multiplicative Analogue of the Normal Distribution

Abstract

We obtain a formula for the density of the free convolution of an arbitrary probability measure on the unit circle of C with the free multiplicative analogues of the normal distribution on the unit circle. This description relies on a characterization of the image of the unit disc under the subordination function, which also allows us to prove some regularity properties of the measures obtained in this way. As an application, we give a new proof for Biane's classic result on the densities of the free multiplicative analogue of the normal distributions. We obtain analogue results for probability measures on R + . Finally, we describe the density of the free multiplicative analogue of the normal distributions as an example and prove unimodality and some symmetry properties of these measures.

Local limit theorems for multiplicative free convolutions

“…Finally, the function η λ t maps the complementary arc {e iθ : θ 1 ≤ |θ| ≤ π} onto the arc {e iθ : θ 0 ≤ |θ| ≤ π} such that η λ t (e iθ 1 ) = e iθ 0 , η λ t (e −iθ 1 ) = e −iθ 0 , and η λ t (−1) = −1. We refer to [24,25] for the details of these calculations. Fix t > 0.…”

“…(Note that the case of approximating free multiplicative Brownian motion on R + by large Brownian motion on the general linear group is proved recently in [13,19].) Free unitary Brownian motion is related to Voiculescu's liberation process in his theory of free Fisher information and free entropy [20], and many of its properties have been intensively studied in the literature [12,24,25].…”

Heat Kernel Empirical Laws on $${\mathbb {U}}_N$$ U N and $${\mathbb {GL}}_N$$ GL N

“…[10] for the t > 0 case and [41] for the t < 0 case. Section 2.5 has further discussion of the measures ν t and their relevance to free probability theory.…”

“…open set in η μ (iC + )). The state of the art is summarized in the following proposition, where the t > 0 statements were proved in [9], while the t < 0 case follows from results in [5][6][7] and the recent preprint [41].…”

Large time unimodality for classical and free Brownian motions with initial distributions

“…The operation ⊠ is called free multiplicative convolution. It can be defined on the set of general probability measures on T. In [26], Zhong proved that for an arbitrary probability measure µ on T the probability measure µ ⊠ λ t is Haar absolutely continuous and its probability density function is continuous on T and is analytic wherever it is positive. In this paper, we study the unimodality for µ ⊠ λ t by using Zhong's density formula and we conclude the following results: Theorems 5.10,5.14,5.15 and 5.17 below.…”