On Asymptotic Growth of the Support of Free Multiplicative Convolutions

Let μ be a compactly supported probability measure on R with expectation 1 and variance V. Let μn denote the n-time free multiplicative convolution of measure μ with itself. Then, for large n the length of the support of μn is asymptotically equivalent to eV n, where e is the base of natural logarithms, e = 2.71 . . . 1 Preliminaries and the main result First, let us recall the definition of the free multiplicative convolution. Let ak denote the moments of a compactly-supported probability measure μ, ak = ∫ tdμ, and let the ψ-transform of μ be ψμ (z) = ∑ ∞ k=1 akz . The inverse ψ-transform is defined as the functional inverse of ψμ (z) and denoted as ψ (−1) μ (z) . It is a well-defined analytic function in a neighborhood of z = 0, provided that a1 6= 0. Suppose that μ and ν are two probability measures supported on R = {x|x ≥ 0} and let ψ (−1) μ (z) and ψ (−1) ν (z) be their inverse ψ-transforms. Then, as it was first shown by Voiculescu in [5], the function f (z) := ( 1 + z ) ψ μ (z) ψ (−1) ν (z) is the inverse ψ-transform of a probability measure supported on R. (Voiculescu used a variant of the inverse ψ-transform, the S-transform.) This new probability measure is called the free multiplicative convolution of measures μ and ν, and denoted as μ ⊠ ν. The significance of this convolution operation can be seen from the fact that if μ and ν are the distributions of singular values of two free operators X and Y, then μ ⊠ ν is the distribution of singular values of the product operator XY (assuming that the algebra containing X and Y is tracial). For more details about free convolutions and free probability theory, the reader can consult [2], [4], or [6]. We are interested in the support of the n-time free multiplicative convolution of the measure