Real Roots of Hypergeometric Polynomials via Finite Free Convolution

In this paper, we prove a result linking the square and the rectangular R-transforms, the consequence of which is a surprising relation between the square and rectangular versions the free additive convolutions, involving the Marchenko–Pastur law. Consequences on random matrices, on infinite divisibility and on the arithmetics of the square versions of the free additive and multiplicative convolutions are given.

The phenomenon of superconvergence, first observed in the central limit theorem of free probability, was subsequently extended to arbitrary limit laws for free additive convolution. We show that the same phenomenon occurs for the multiplicative versions of free convolution on the positive line and on the unit circle. We also show that a certain Hölder regularity, first demonstrated by Biane for the density of a free additive convolution with a semicircular law, extends to free (additive and multiplicative) convolutions with arbitrary freely infinitely divisible distributions.

Free additive convolution is a binary operation on the set M of all probability measures on the real line R. This operation was first defined in [7] for measures with finite moments of all orders (in particular for compactly supported measures). Maassen [5] extended this operation to measures with finite variance, and the extension to arbitrary measures was done in [1]. The free convolution of µ,v ∈ M is denoted µ ⊞ v. Unlike classical convolution, free convolution is a highly nonlinear operation, and therefore it is not obvious that various regularity properties of µ (like absolute continuity, differentiability, etc.) should be passed on to µ ⊞ v. In some respects however free convolution has a stronger regularizing effect than classical convolution. It is our purpose in this paper to examine a few instances in which regularity properties of µ ⊞ v can be inferred. Some of the earlier results in this direction were only proved for measures with compact support. We will extend these results to general probability measures, giving as much of the technical detail as necessary. Among the new results, we show that there may be a loss of smoothness under free convolution. We also give a complete description of the atoms of a free convolution of probability measures.KeywordsCompact SupportSelfadjoint OperatorAbsolute ContinuityFree ConvolutionUnbounded SupportThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We give an analytical approach to the definition of additive and multiplicative free convolutions which is based on the theory of Nevanlinna and Schur functions. We consider the set of probability distributions as a semigroup M equipped with the operation of free convolution and prove a Khintchine type theorem for the factorization of elements of this semigroup. An element of M contains either indecomposable (“prime”) factors or it belongs to a class, say I 0, of distributions without indecomposable factors. In contrast to the classical convolution semigroup, in the free additive and multiplicative convolution semigroups the class I 0 consists of units (i.e. Dirac measures) only. Furthermore we show that the set of indecomposable elements is dense in M.

In his article "On the free convolution with a semicircular distribution,"\nBiane found very useful characterizations of the boundary values of the\nimaginary part of the Cauchy-Stieltjes transform of the free additive\nconvolution of a probability measure on the real line with a Wigner\n(semicircular) distribution. Biane's methods were recently extended by Huang to\nmeasures which belong to the partial free convolution semigroups introduced by\nNica and Speicher. This note further extends some of Biane's methods and\nresults to free convolution powers of operator-valued distributions and to free\nconvolutions with operator-valued semicirculars. In addition, it investigates\nproperties of the Julia-Caratheodory derivative of the subordination functions\nassociated to such semigroups, extending certain results from the article\n"Partially Defined Semigroups Relative to Multiplicative Free Convolution" by\nBercovici and the author (reference [7] in the text).\n

It has been shown by Voiculescu and Biane that the analytic subordina- tion property holds for free additive and multiplicative convolutions. In this paper, we present an operatorial approach to subordination for free multiplicative convolutions. This study is based on the concepts of 'freeness with subordination', or 's-free inde- pendence', and 'orthogonal independence', introduced recently in the context of free additive convolutions. In particular, we introduce and study the associated multiplica- tive convolutions and construct related operators, called 'subordination operators' and 'subordination branches'. Using orthogonal independence, we derive decompositions of subordination branches and related decompositions of s-free and free multiplicative convolutions. The operatorial methods lead to several new types of graph products, called 'loop products', associated with different notions of independence (monotone, boolean, orthogonal, s-free). We also prove that the enumeration of rooted 'alternat- ing double return walks' on the loop products of graphs and on the free product of graphs gives the moments of the corresponding multiplicative convolutions.

Let F(νj)={Qmjνj,mj∈(m−νj,m+νj)}, j=1,2, be two Cauchy–Stieltjes Kernel (CSK) families induced by non-degenerate compactly supported probability measures ν1 and ν2. Introduce the set of measures F=F(ν1)⊞F(ν2)={Qm1ν1⊞Qm2ν2,m1∈(m−ν1,m+ν1)andm2∈(m−ν2,m+ν2)}. We show that if F remains a CSK family, (i.e., F=F(μ) where μ is a non-degenerate compactly supported measure), then the measures μ, ν1 and ν2 are of the Marchenko–Pastur type measure up to affinity. A similar conclusion is obtained if we substitute (in the definition of F) the additive free convolution ⊞ by the additive Boolean convolution ⊎. The cases where the additive free convolution ⊞ is replaced (in the definition of F) by the multiplicative free convolution ⊠ or the multiplicative Boolean convolution ⨃ are also studied.

Relations between convolutions and transforms in operator-valued free probability

This work proposes algorithms for computing additive and multiplicative free convolutions of two given probability measures. We consider probability measures with compact support whose free convolution results in a probability measure with a density function that exhibits a square-root decay at the boundary (for example, the semicircle distribution or the Marchenko–Pastur distribution). A key ingredient of our method is rewriting the intermediate quantities of the free convolution using the Cauchy integral formula and then discretizing these integrals using the trapezoidal quadrature rule, which converges exponentially fast under suitable analyticity properties of the functions to be integrated.

In this thesis we study the role of k-divisible non-crossing partitions in Free Probability. First, we consider the combinatorial convolution ∗ in the lattices NC of non-crossing partitions and NC of k-divisible non-crossing partitions. We show that convolving k times with the zeta-function in NC is equivalent to convolving once with the zeta-function in NC. This gives new ways of counting objects like k-equal partitions, k-divisible partitions and k-multichains both in NC and NC. We also consider some statistics of block sizes in k-divisible non-crossing partitions. Second, we introduce and study the notion of k-divisible elements in a non-commutative probability space. A k-divisible element is a (non-commutative) random variable whose n-th moment vanishes whenever n is not a multiple of k. For such k-divisible element x, we derive a formula for the free cumulants of x in terms of the free cumulants of x. For this we use our combinatorial results on the lattice of k-divisible non-crossing partitions. We prove that if a and s are free and s is k-divisible then sps and a are free, where p is any polynomial (in a and s) of degree k − 2 in s. Moreover, we define a notion of R-diagonal k-tuples and prove similar results. Next, we show that free multiplicative convolution between a measure concentrated on the positive real line and a probability measure with k-symmetry is well defined. Analytic tools to calculate this convolution are developed. We then concentrate on free additive powers of k-symmetric distributions and prove that μ t is a well defined probability measure, for all t > 1. We derive central limit theorems and Poisson type ones. More generally, we consider freely infinitely divisible measures and prove that free infinite divisibility is maintained under the mapping μ→ μ. Relations between free multiplicative powers and k-divisible non-crossing partitions are also found and generalized to any product of free random variables. We conclude by focusing on (k-symmetric) free stable distributions, for which we prove a reproducing property generalizing the ones known for one sided and real symmetric free stable laws.

The investigation of free additive convolution is a key concept in free probability theory, offering a framework for studying the sum of freely independent random variables. This paper uses free additive convolution and measure dilations to investigate various aspects of Marchenko–Pastur and free Gamma laws in the setting of Cauchy-Stieltjes Kernel (CSK) families. Our investigation reveals the essential links between analytic function theory and free probability, highlighting the usefulness of CSK families in developing the theoretical and computational aspects of free additive convolution.

A notion of generalized (two-parameterized) t-transformation of free convolution, also called (t=(a,b))-deformed free convolution, is introduced for a∈R and b>0. In this article, some results of t-deformed free convolution are given within the theory of Cauchy-Stieltjes Kernel (CSK) families. The variance function is a fundamental concept in CSK families. An expression is provided for the variance function under t-deformed free convolution power. In addition, through the use of the variance function, an approximation is provided for members of the t-deformed free Gaussian CSK family and members of the t-deformed free Poisson CSK family respectively. Furthermore, by involving the free multiplicative convolution, a new limit theorem is provided with respect to t-deformed free convolution.

We investigate the level density for several ensembles of positive random matrices of a Wishart-like structure, W=XX(†), where X stands for a non-Hermitian random matrix. In particular, making use of the Cauchy transform, we study the free multiplicative powers of the Marchenko-Pastur (MP) distribution, MP(⊠s), which for an integer s yield Fuss-Catalan distributions corresponding to a product of s-independent square random matrices, X=X(1)⋯X(s). New formulas for the level densities are derived for s=3 and s=1/3. Moreover, the level density corresponding to the generalized Bures distribution, given by the free convolution of arcsine and MP distributions, is obtained. We also explain the reason of such a curious convolution. The technique proposed here allows for the derivation of the level densities for several other cases.

Subordination is the basis of the analytic approach to free additive and multiplicative convolution. We extend this approach to a more general setting and prove that the conditional expectation [Formula: see text] for free random variables [Formula: see text] and a Borel function [Formula: see text] is a resolvent again. This result allows the explicit calculation of the distribution of noncommutative polynomials of the form [Formula: see text]. The main tool is a new combinatorial formula for conditional expectations in terms of Boolean cumulants and a corresponding analytic formula for conditional expectations of resolvents, generalizing subordination formulas for both additive and multiplicative free convolutions. In the final section, we illustrate the results with step by step explicit computations and an exposition of all necessary ingredients.

In classical probability the law of large numbers for the multiplicative convolution follows directly from the law for the additive convolution. In free probability this is not the case. The free additive law was proved by D. Voiculescu in 1986 for probability measures with bounded support and extended to all probability measures with first moment by J. M. Lindsay and V. Pata in 1997, while the free multiplicative law was proved only recently by G. Tucci in 2010. In this paper we extend Tucci's result to measures with unbounded support while at the same time giving a more elementary proof for the case of bounded support. In contrast to the classical multiplicative convolution case, the limit measure for the free multiplicative law of large numbers is not a Dirac measure, unless the original measure is a Dirac measure. We also show that the mean value of \ln x is additive with respect to the free multiplicative convolution while the variance of \ln x is not in general additive. Furthermore we study the two parameter family (\mu_{\alpha,\beta})_{\alpha,\beta \ge 0} of measures on (0,\infty) for which the S-transform is given by S_{\mu_{\alpha,\beta}}(z) = (-z)^\beta (1+z)^{-\alpha}, 0 < z < 1.