The normal distribution is ⊞-infinitely divisible

Abstract

We prove that the classical normal distribution is infinitely divisible with respect to the free additive convolution. We study the Voiculescu transform first by giving a survey of its combinatorial implications and then analytically, including a proof of free infinite divisibility. In fact we prove that a subfamily Askey-Wimp-Kerov distributions are freely infinitely divisible, of which the normal distribution is a special case.The Cauchy-Stieltjes transform of a measure µ on the real line is defined bywhen µ is a positive measure, this function maps C + into the lower half-plane. For details we refer to the excellent book of Akhieser [1]. For c = 0, we have µ 0 = γ, the normal distribution [27], and one can easily check that one can extend this family continuously to c = −1 by letting µ −1 = δ 0 , the probability giving mass one to {0}. This family, introduced in [4], plays an important role in [27]; we shall call its members the Askey-Wimp-Kerov distributions. It will turn out from our proof that {µ c : c ∈ [−1, 0]} are freely infinitely divisible. Numerical computations show that for several values of c > 0, µ c is not freely infinitely divisible. Numerical evidence seems also to indicate that µ c is classicaly infinitely divisible only when c = 0 or c = −1. An interesting interpolation between the normal and the semicircle law was constructed by Bryc, Dembo and Jiang [18] and further investigated by Buchholz [19]. This leads to a generalized Brownian motion, given by a weight function (0 < b < 1)

On the non-classical infinite divisibility of power semicircle distributions

“…This fact is also supported by using MATLAB for testing the positive definiteness of the free cumulants of the Gaussian distribution. This conjecture has recently been proved to be true in [4]. Moreover, we conjecture that the classical Gaussian distribution is a free multiplicative convolution of the semicircle distribution in the sense of [3], [40].…”

“…We also include results and conjectures on the free infinite divisibility of the classical Gaussian distribution, a result recently proved in Belinschi et al [4]. In particular, we show that the free divisibility indicator of the classical Gaussian distribution is strictly less than 2.…”

Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem

“…The function ω(i ) is specified in [3], however its exact form is not necessary for our purpose, since the upper diagonal element of the Green's function satisfies the relation [3, eq. (32)]…”

“…Relevance of the correlation function and its connection with the eigenvalue condition number is discussed in Section 3. Section 4 exploits the formalism and results of [3], in order to provide a short, direct proof of the main result (1). Section 5 includes few examples where our formula can be easily applied and provides the verification of these results with the large scale numerical simulations.…”

“…Recently, Belinschi, Speicher andŚniady [3] provided a rigorous mathematical justification of the method of the so-called generalized Green's functions [1,4,5,6,7], broadly exploited in the physics literature in relation to the spectral problems of non-hermitian operators. In particular, as one of the examples of their construction, they have shown explicitly, how the generalized Green's functions reproduce the HaagerupLarsen theorem [2], valid for the case when the non-hermitian operator X can be decomposed as X = P U , where P is a positive hermitian operator, U is a Haar unitary operator and P and U are mutually free in the sense of free random variables [8].…”