Free Additive Convolution Research Articles

Density of the Free Additive Convolution of Multi-cut Measures

Abstract We consider the free additive convolution semigroup $\lbrace \mu ^{\boxplus t}:\,t\ge 1\rbrace $ and determine the local behavior of the density of $\mu ^{\boxplus t}$ at the endpoints and at any singular point of its support. We then study the free additive convolution of two multi-cut probability measures and show that its density decays either as a square root or as a cubic root at any endpoint of its support. The probability measures considered in this paper satisfy a power law behavior with exponents strictly between $-1$ and $1$ at the endpoints of their supports.

A unified approach to infinitesimal freeness with amalgamation

In this paper, we consider the infinitesimal freeness in the operator-valued framework, and we show that the operator-valued infinitesimal (OVI) free independence is equivalent to the operator-valued free independence over an algebra of [Formula: see text] upper triangular matrices. We introduce the notion of OVI cumulants and investigate its properties, and we then deduce that the OVI freeness is equivalent to the vanishing of our mixed cumulants. Moreover, we derive the formula for obtaining the free additive and multiplicative convolutions within the realm of OVI freeness.

Regularity for free multiplicative convolution on the unit circle

It is shown that the free multiplicative convolution of two nondegenerate probability measures on the unit circle has no continuous singular part relative to arclength measure. Analogous results have long been known for free additive convolutions on the line and free multiplicative convolution on the positive half-line.

Superconvergence and regularity of densities in free probability

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.

Brown measures of free circular and multiplicative Brownian motions with self-adjoint and unitary initial conditions

Let Z_N be a Ginibre ensemble and let A_N be a Hermitian random matrix independent of Z_N such that A_N converges in distribution to a self-adjoint random variable x_0 in a W^* -probability space (\mathscr{A},\tau) . For each t>0 , the random matrix A_N+\sqrt{t}\,Z_N converges in \ast -distribution to x_0+c_t , where c_t is a circular variable of variance t , freely independent of x_0 . We use the Hamilton–Jacobi method to compute the Brown measure \rho_t of x_0+c_t . The Brown measure has a density that is constant along the vertical direction inside the support. The support of the Brown measure of x_0+c_t is related to the subordination function of the free additive convolution of x_0+s_t , where s_t is a semicircular variable of variance t , freely independent of x_0 . Furthermore, the push-forward of \rho_t by a natural map is the law of x_0+s_t . Let G_N(t) be the Brownian motion on the general linear group and let U_N be a unitary random matrix independent of G_N(t) such that U_N converges in distribution to a unitary random variable u in (\mathscr{A},\tau) . The random matrix U_NG_N(t) converges in \ast -distribution to ub_t where b_t is the free multiplicative Brownian motion, freely independent of u . We compute the Brown measure \mu_t of ub_t , extending the recent work by Driver–Hall–Kemp, which corresponds to the case u=I . The measure has a density of the special form \frac{1}{r^2}w_t(\theta) in polar coordinates in its support. The support of \mu_t is related to the subordination function of the free multiplicative convolution of uu_t where u_t is the free unitary Brownian motion, freely independent of u . The push-forward of \mu_t by a natural map is the law of uu_t . In the special case that u is Haar unitary, the Brown measure \mu_t follows the annulus law . The support of the Brown measure of ub_t is an annulus with inner radius e^{-t/2} and outer radius e^{t/2} . In its support, the density in polar coordinates is given by \frac{1}{2\pi t}\,\frac{1}{r^2}.

Regularity results for free Lévy processes

Given a free additive convolution semigroup (μt)t≥0 and a probability measure ν on R, we find the necessary and sufficient conditions for the process μt⊞ν to be Lebesgue absolutely continuous with a positive and analytic density throughout R at all time t>0. For semigroups without this property, we find the necessary and sufficient conditions for the density of μt⊞ν to be analytic at its zeros. These results are quantified by the Lévy measure of the semigroup, making it fairly easy to construct many concrete examples. Finally, we show that μt⊞ν has a finite number of connected components in its support if both the Lévy measure of (μt)t≥0 and the initial law ν do.

Principal angles between random subspaces and polynomials in two free projections

We use the geometric concept of principal angles between subspaces to compute the noncommutative distribution of an expression involving two free projections. For example, this allows to simplify a formula by Fevrier-Mastnak-Nica-Szpojankowski about the free Bernoulli anticommutator. We also derive economically an explicit formula for the free additive convolution of Bernoulli distributions. As a byproduct, we observe the remarkable fact that the principal angles between random half-dimensional subspaces are asymptotically distributed according to the uniform measure.

Some results in Cauchy-Stieltjes kernel families

In this paper we present two different results in the theory of Cauchy-Stieltjes Kernel (CSK) families. We firstly provide the construction of free Sheffer systems with the theory of CSK families. We associate a free additive convolution semigroup of probability measures to any free Sheffer systems and we prove that this is the only one that leads to an orthogonal free Sheffer systems. We also show that the orthogonality of free Sheffer systems occurs if and only if the associated free additive convolution semigroup of probability measures generates CSK families with quadratic variance function. Secondly, we are interested in the study of boolean additive convolution. Based on the criteria of convergence for a sequence of variance functions we give an approximation of elements of the CSK family generated by the boolean Gaussian distribution and an approximation of elements of the CSK family generated by the boolean Poisson distribution.

Relations between convolutions and transforms in operator-valued free probability

We introduce a class of independence relations, which include free, boolean and monotone independence, in operator-valued probability. We show that this class of independence relations have a matricial extension property so that we can easily study their associated convolutions via Voiculescu's fully matricial function theory. Based the matricial extension property, we show that many results can be generalized to multi-variable cases. Besides free, boolean and monotone additive convolutions, we will focus on two other important additive convolutions, which are orthogonal and subordination additive convolutions. We show that the operator-valued subordination functions, which come from the free additive convolutions or the operator-valued free convolution powers, are reciprocal Cauchy transforms of operator-valued random variables which are uniquely determined up to Voiculescu's fully matricial function theory. In the end, we study relations between certain convolutions and transforms in C⁎-operator-valued probability.

Central limit theorem for mesoscopic eigenvalue statistics of deformed Wigner matrices and sample covariance matrices

We consider N by N deformed Wigner random matrices of the form XN=HN+AN, where HN is a real symmetric or complex Hermitian Wigner matrix and AN is a deterministic real bounded diagonal matrix. We prove a universal Central Limit Theorem for the linear eigenvalue statistics of XN for all mesoscopic scales both in the spectral bulk and at regular edges where the global eigenvalue density vanishes as a square root. The method relies on studying the characteristic function of the linear statistics (Landon and Sosoe (2018)) by using the cumulant expansion method, along with local laws for the Green function of XN (Ann. Probab. 48 (2020) 963–1001; Probab. Theory Related Fields 169 (2017) 257–352; J. Math. Phys. 54 (2013) 103504) and analytic subordination properties of the free additive convolution (Dallaporta and Fevrier (2019); Random Matrices Theory Appl. 9 (2020) 2050011). We also prove the analogous results for high-dimensional sample covariance matrices.

Boolean cumulants and subordination in free probability

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.

On the support of the free additive convolution

We consider the free additive convolution of two probability measures μ and ν on the real line and show that μ ⊞ v is supported on a single interval if μ and ν each has single interval support. Moreover, the density of μ ⊞ ν is proven to vanish as a square root near the edges of its support if both μ and ν have power law behavior with exponents between −1 and 1 near their edges. In particular, these results show the ubiquity of the conditions in our recent work on optimal local law at the spectral edges for addition of random matrices [5].

Sums of random polynomials with independent roots

We consider the zeros of the sum of independent random polynomials as their degrees tend to infinity. Namely, let p and q be two independent random polynomials of degree n, whose roots are chosen independently from the probability measures μ and ν in the complex plane, respectively. We compute the limiting distribution for the zeros of the sum p+q as n tends to infinity. The limiting distribution can be described by its logarithmic potential, which we show is the pointwise maximum of the logarithmic potentials of μ and ν. More generally, we consider the sum of m independent degree n random polynomials when m is fixed and n tends to infinity. Our results can be viewed as describing a version of the free additive convolution from free probability theory for zeros of polynomials.

Central Limit Theorem for Mesoscopic Eigenvalue Statistics of the Free Sum of Matrices

Abstract We consider random matrices of the form $H_N=A_N+U_N B_N U^*_N$, where $A_N$ and $B_N$ are two $N$ by $N$ deterministic Hermitian matrices and $U_N$ is a Haar distributed random unitary matrix. We establish a universal central limit theorem for the linear eigenvalue statistics of $H_N$ on all mesoscopic scales inside the regular bulk of the spectrum. The proof is based on studying the characteristic function of the linear eigenvalue statistics and consists of two main steps: (1) generating Ward identities using the left-translation invariance of the Haar measure, along with a local law for the resolvent of $H_N$ and analytic subordination properties of the free additive convolution, allows us to derive an explicit formula for the derivative of the characteristic function; (2) a local law for two-point product functions of resolvents is derived using a partial randomness decomposition of the Haar measure. We also prove the corresponding results for orthogonal conjugations.

Spectral rigidity for addition of random matrices at the regular edge

We consider the sum of two large Hermitian matrices A and B with a Haar unitary conjugation bringing them into a general relative position. We prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is asymptotically given by the free additive convolution of the laws of A and B as the dimension of the matrix increases. This implies optimal rigidity of the eigenvalues and optimal rate of convergence in Voiculescu's theorem. Our previous works [4,5] established these results in the bulk spectrum, the current paper completely settles the problem at the spectral edges provided they have the typical square-root behavior. The key element of our proof is to compensate the deterioration of the stability of the subordination equations by sharp error estimates that properly account for the local density near the edge. Our results also hold if the Haar unitary matrix is replaced by the Haar orthogonal matrix.

A construction which relates c-freeness to infinitesimal freeness

We consider two extensions of free probability that have been studied in the research literature, and are based on the notions of c-freeness and respectively of infinitesimal freeness for noncommutative random variables. In a 2012 paper, Belinschi and Shlyakhtenko pointed out a connection between these two frameworks, at the level of their operations of 1-dimensional free additive convolution. Motivated by that, we propose a construction which produces a multi-variate version of the Belinschi-Shlyakhtenko result, together with a result concerning free products of multi-variate noncommutative distributions. Our arguments are based on the combinatorics of the specific types of cumulants used in c-free and in infinitesimal free probability. They work in a rather general setting, where the initial data consists of a vector space V given together with a linear map Δ:V→V⊗V. In this setting, all the needed brands of cumulants live in the guise of families of multilinear functionals on V, and our main result concerns a certain transformation Δ⁎ on such families of multilinear functionals.

Generalised operations in free harmonic analysis

In this article we study the formal side of operations in free harmonic analysis and examine the emerging general picture of all this. We establish an analytic correspondence of semi-rings between Witt vectors and free probability, by building on previous joint work with Friedrich and McKay (Formal groups, Witt vectors and free probability, 2012. arXiv:1204.6522). In particular, an exponential map, which relates the free additive convolution semigroup on $${\mathbb {R}}$$ with the free multiplicative convolution semigroup on either the unit circle or the positive real axis of compactly supported, freely infinitely divisible probability measures, is derived with complex analytic methods. Then we define several novel operations on these sets, discuss their relation with classically infinitely divisible measures and determine the internal geometry of the spaces involved. Finally, we formalise the structure induced by the various operations we have introduced, in the language of operads and algebraic theories.

Eta-diagonal distributions and infinite divisibility for R-diagonals

Dans le cadre des probabilités libres, la classe des $*$-distributions $R$-diagonales est assez bien comprise. Dans cette classe, nous nous intéressons au concept d’infinie divisibilité par rapport à l’opération de convolution additive libre $\boxplus$. Nous exploitons la relation entre probabilités libres et le monde parallèle (et plus simple) des probabilités booléennes. Il est naturel d’introduire le concept de distributions $\eta$-diagonales, qui sont la contrepartie booléenne des distributions $R$-diagonales. Nous établissons un certain nombre de propriétés des distributions $\eta$-diagonales, avant d’examiner la bijection canonique reliant les distributions $\eta$-diagonales aux distributions $R$-diagonales indéfiniment divisibles. Le résultat principal est la paramétrisation par une paire de mesures boreliennes sur $[0,\infty)$ à support compact de toutes les lois $R$-diagonales $\boxplus$-indéfiniment divisibles pouvant apparaître dans un $C^{*}$-espace de probabilités. Parmi les applications de cette paramétrisation, nous montrons que l’ensemble des distributions $R$-diagonales $\boxplus$-indéfiniment divisibles est fermé sous l’opération $\boxtimes$ de convolution multiplicative libre.

Analytic subordination for bi-free convolution

In this paper we study some analytic properties of bi-free additive convolution, both scalar- and operator-valued. We show that using properties of Voiculescu's subordination functions associated to free additive convolution of operator-valued distributions, simpler formulas for bi-free convolutions can be derived. We use these formulas in order to prove several results about atoms of bi-free additive convolutions.

Convergence rate for spectral distribution of addition of random matrices

Let A and B be two N by N deterministic Hermitian matrices and let U be an N by N Haar distributed unitary matrix. It is well known that the spectral distribution of the sum H=A+UBU⁎ converges weakly to the free additive convolution of the spectral distributions of A and B, as N tends to infinity. We establish the optimal convergence rate 1N in the bulk of the spectrum.