Noncommutative Random Variables Research Articles

Markovianity and the Thompson monoid F+

We introduce a new distributional invariance principle, called ‘partial spreadability’, which emerges from the representation theory of the Thompson monoid F+ in noncommutative probability spaces. We show that a partially spreadable sequence of noncommutative random variables is adapted to a local Markov filtration. Conversely we show that a large class of noncommutative stationary Markov sequences provides representations of the Thompson monoid F+. In the particular case of a classical probability space, we arrive at a de Finetti theorem for stationary Markov sequences with values in a standard Borel space.

Free moment measures and laws

In [3], it was shown that convex, almost everywhere continuous functions coordinatize a broad class of probability measures on Rn by the map U↦(∇U)#e−Udx. We consider whether there is a similar coordinatization of non-commutative probability spaces, with the Gibbs measure e−Udx replaced by the corresponding free Gibbs law. We call laws parameterized in this way free moment laws. We first consider the case of a single (and thus commutative) random variable and then the regime of n non-commutative random variables which are perturbations of freely independent semi-circular variables. We prove that free moment laws exist with little restriction for the one dimensional case, and for small even perturbations of free semi-circle laws in the general case.

Duality for Optimal Couplings in Free Probability

We study the free probabilistic analog of optimal couplings for the quadratic cost, where classical probability spaces are replaced by tracial von Neumann algebras, and probability measures on {mathbb {R}}^m are replaced by non-commutative laws of m-tuples. We prove an analog of the Monge–Kantorovich duality which characterizes optimal couplings of non-commutative laws with respect to Biane and Voiculescu’s non-commutative L^2-Wasserstein distance using a new type of convex functions. As a consequence, we show that if (X, Y) is a pair of optimally coupled m-tuples of non-commutative random variables in a tracial mathrm {W}^*-algebra mathcal {A}, then mathrm {W}^*((1 - t)X + tY) = mathrm {W}^*(X,Y) for all t in (0,1). Finally, we illustrate the subtleties of non-commutative optimal couplings through connections with results in quantum information theory and operator algebras. For instance, two non-commutative laws that can be realized in finite-dimensional algebras may still require an infinite-dimensional algebra to optimally couple. Moreover, the space of non-commutative laws of m-tuples is not separable with respect to the Wasserstein distance for m > 1.

Rates of convergence for laws of the spectral maximum of free random variables

Let [Formula: see text] be a sequence of freely independent, identically distributed non-commutative random variables. Consider a sequence [Formula: see text] of the renormalized spectral maximum of random variables [Formula: see text]. It is known that the renormalized spectral maximum [Formula: see text] converges to the free extreme value distribution under certain conditions on the distribution function. In this paper, we provide a rate of convergence in the Kolmogorov distance between a distribution function of [Formula: see text] and the free extreme value distribution.

Free CIR processes

For stochastic processes of non-commuting random variables, we formulate a Cox–Ingersoll–Ross (CIR) stochastic differential equation in the context of free probability theory which was introduced by D. Voiculescu. By transforming the classical CIR equation and the Feller condition, which ensures the existence of a positive solution, into the free setting (in the sense of having a strictly positive spectrum), we show the global existence for a free CIR equation. The main challenge lies in the transition from a stochastic differential equation driven by a classical Brownian motion to a stochastic differential equation driven by the free analogue to the classical Brownian motion, the so-called free Brownian motion.

Regression conditions that characterize free-Poisson and free-Kummer distributions

We find the asymptotic spectral distribution of random Kummer matrix. Then we formulate and prove a free analogue of HV independence property, which is known for classical Kummer and Gamma random variables and for Kummer and Wishart matrices. We also prove a related characterization of free-Kummer and free-Poisson (Marchenko–Pastur) non-commutative random variables.

Maximal correlation and monotonicity of free entropy and of Stein discrepancy

We introduce the maximal correlation coefficient R(M1,M2) between two noncommutative probability subspaces M1and M2 and show that the maximal correlation coefficient between the sub-algebras generated by sn:=x1+…+xnand sm:=x1+…+xm equals m∕n for m≤n, where (xi)i∈N is a sequence of free and identically distributed noncommutative random variables. This is the free-probability analogue of a result by Dembo–Kagan–Shepp in classical probability. As an application, we use this estimate to provide another simple proof of the monotonicity of the free entropy and free Fisher information in the free central limit theorem. Moreover, we prove that the free Stein Discrepancy introduced by Fathi and Nelson is non-increasing along the free central limit theorem.

Random Matrices

Large complex systems tend to develop universal patterns that often represent their essential characteristics. For example, the cumulative effects of independent or weakly dependent random variables often yield the Gaussian universality class via the central limit theorem. For non-commutative random variables, e.g. matrices, the Gaussian behavior is often replaced by another universality class, commonly called random matrix statistics. Nearby eigenvalues are strongly correlated, and, remarkably, their correlation structure is universal, depending only on the symmetry type of the matrix. Even more surprisingly, this feature is not restricted to matrices; in fact Eugene Wigner, the pioneer of the field, discovered in the 1950s that distributions of the gaps between energy levels of complicated quantum systems universally follow the same random matrix statistics. This claim has never been rigorously proved for any realistic physical system but experimental data and extensive numerics leave no doubt as to its correctness. Since then random matrices have proved to be extremely useful phenomenological models in a wide range of applications beyond quantum physics that include number theory, statistics, neuroscience, population dynamics, wireless communication and mathematical finance. The ubiquity of random matrices in natural sciences is still a mystery, but recent years have witnessed a breakthrough in the mathematical description of the statistical structure of their spectrum. Random matrices and closely related areas such as log-gases have become an extremely active research area in probability theory. This workshop brought together outstanding researchers from a variety of mathematical backgrounds whose areas of research are linked to random matrices. While there are strong links between their motivations, the techniques used by these researchers span a large swath of mathematics, ranging from purely algebraic techniques to stochastic analysis, classical probability theory, operator algebra, supersymmetry, orthogonal polynomials, etc.

A Central Limit Theorem for Star-Generators of $${S}_{\infty }$$, Which Relates to the Law of a GUE Matrix

It is well known that, on a purely algebraic level, a simplified version of the central limit theorem (CLT) can be proved in the framework of a non-commutative probability space, under the hypotheses that the sequence of non-commutative random variables we consider is exchangeable and obeys a certain vanishing condition of some of its joint moments. In this approach (which covers versions for both the classical CLT and the CLT of free probability), the determination of the resulting limit law has to be addressed on a case-by-case basis. In this paper we discuss an instance of the above theorem that takes place in the framework of the group algebra $${{\mathbb {C}}}[ S_{\infty } ]$$ of the infinite symmetric group: The exchangeable sequence is provided by the star-generators of $$S_{\infty }$$ , and the expectation functional used on $${{\mathbb {C}}}[ S_{\infty } ]$$ depends in a natural way on a parameter $$d \in {{\mathbb {N}}}$$ . We identify precisely the limit distribution $$\mu _d$$ for this special instance of CLT, via a connection that $$\mu _d$$ turns out to have with the average empirical eigenvalue distribution of a random $$d \times d$$ GUE matrix. Moreover, we put into evidence a multivariate version of this result which follows from the observation that, on the level of calculations with pair-partitions, the (non-centered) star-generators are related to a (centered) exchangeable sequence of GUE matrices with independent entries.

Conditional Expectation, Entropy, and Transport for Convex Gibbs Laws in Free Probability

Abstract Let $(X_1,\dots ,X_m)$ be self-adjoint noncommutative random variables distributed according to the free Gibbs law given by a sufficiently regular convex and semi-concave potential $V$, and let $(S_1,\dots ,S_m)$ be a free semicircular family. For $k < m$, we show that conditional expectations and conditional non-microstates free entropy given $X_1$, …, $X_k$ arise as the large $N$ limit of the corresponding conditional expectations and entropy for the $N \times N$ random matrix models associated to $V$. Then, by studying conditional transport of measure for the matrix models, we construct an isomorphism $\mathrm{W}^*(X_1,\dots ,X_m) \to \mathrm{W}^*(S_1,\dots ,S_m)$ that maps $\mathrm{W}^*(X_1,\dots ,X_k)$ to $\mathrm{W}^*(S_1,\dots ,S_k)$ for each $k = 1, \dots , m$ and that also witnesses the Talagrand inequality for the law of $(X_1,\dots ,X_m)$ relative to the law of $(S_1,\dots ,S_m)$.

Hölder continuity of cumulative distribution functions for noncommutative polynomials under finite free Fisher information

This paper contributes to the current studies on regularity properties of noncommutative distributions in free probability theory. More precisely, we consider evaluations of selfadjoint noncommutative polynomials in noncommutative random variables that have finite non-microstates free Fisher information, highlighting the special case of Lipschitz conjugate variables. For the first time in this generality, it is shown that the analytic distributions of those evaluations have Hölder continuous cumulative distribution functions with an explicit Hölder exponent that depends only on the degree of the considered polynomial. For linear polynomials, we reach in the case of finite non-microstates free Fisher information the optimal Hölder exponent 23, and get Lipschitz continuity in the case of Lipschitz conjugate variables. In particular, our results guarantee that such polynomial evaluations have finite logarithmic energy and thus finite (non-microstates) free entropy, which partially settles a conjecture of Charlesworth and Shlyakhtenko [8].We further provide a very general criterion that gives for weak approximations of measures having Hölder continuous cumulative distribution functions explicit rates of convergence in terms of the Kolmogorov distance.Finally, we combine these results to study the asymptotic eigenvalue distributions of polynomials in GUEs or matrices with more general Gibbs laws. For Gibbs laws, this extends the corresponding result obtained in [21] from convergence in distribution to convergence in Kolmogorov distance; in the GUE case, we even provide explicit rates, which quantify results of [23,24] in terms of the Kolmogorov distance.

Inequalities for acceptable noncommutative random variables

We introduce the notion of acceptable noncommutative random variables and investigate their essential properties. More precisely, we provide several efficient estimation of tail probabilities of sums of noncommutative random variables under some mild conditions. Moreover, we investigate the complete convergence of a sequence of the form [Formula: see text] in which [Formula: see text] is a sequence of acceptable noncommutative random variables. Our results extend some classical inequalities as well.

Operator-valued chordal Loewner chains and non-commutative probability

We adapt the theory of chordal Loewner chains to the operator-valued matricial upper-half plane over a C⁎-algebra A. We define an A-valued chordal Loewner chain as a subordination chain of analytic self-maps of the A-valued upper half-plane, such that each Ft is the reciprocal Cauchy transform of an A-valued law μt, such that the mean and variance of μt are continuous functions of t.We relate A-valued Loewner chains to processes with A-valued free or monotone independent increments just as was done in the scalar case by Bauer [8] and Schleißinger [48]. We show that the Loewner equation ∂tFt(z)=DFt(z)[Vt(z)], when interpreted in a certain distributional sense, defines a bijection between Lipschitz mean-zero Loewner chains Ft and vector fields Vt(z) of the form Vt(z)=−Gνt(z) where νt is a generalized A-valued law.Based on the Loewner equation, we derive a combinatorial expression for the moments of μt in terms of νt. We also construct non-commutative random variables on an operator-valued monotone Fock space which realize the laws μt. Finally, we prove a version of the monotone central limit theorem which describes the behavior of Ft as t→+∞ when νt has uniformly bounded support.

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.

On the global fluctuations of block Gaussian matrices

In this paper we study the global fluctuations of block Gaussian matrices within the framework of second-order free probability theory. In order to compute the second-order Cauchy transform of these matrices, we introduce a matricial second-order conditional expectation and compute the matricial second-order Cauchy transform of a certain type of non-commutative random variables. As a by-product, using the linearization technique, we obtain the second-order Cauchy transform of non-commutative rational functions evaluated on selfadjoint Gaussian matrices.

Operations on certain non-commutative operator-valued random variables

Operations on certain non-commutative operator-valued random variables

A variance bound for a general function of independent noncommutative random variables

The main purpose of this paper is to establish a noncommutative analogue of the Efron-Stein inequality, which bounds the variance of a general function of some independent random variables. Moreover, we state an operator version including random matrices, which extends a result of D. Paulin et al., [Ann. Probab. 44(5) (2016), 3431–3473]. Further, we state a Steele type inequality in the framework of noncommutative probability spaces.

On the multiplication of operator-valued c-free random variables

We discuss some results concerning the multiplication of non-commutative random variables that are c-free with respect to a pair $( \Phi, \varphi) $, where $ \Phi $ is a linear map with values in some Banach or C$^\ast$-algebra and $ \varphi $ is scalar-valued. In particular, we construct a suitable analogue of the Voiculescu's $ S $-transform for this framework.

Extended de Finetti theorems for boolean independence and monotone independence

We construct several new spaces of quantum sequences and their quantum families of maps in the sense of Sołtan. The noncommutative distributional symmetries associated with these quantum maps are noncommutative versions of spreadability and partial exchangeability. Then, we study simple relations between these symmetries. We will focus on studying two kinds of noncommutative distributional symmetries: monotone spreadability and boolean spreadability. We provide an example of a spreadable sequence of random variables for which the usual unilateral shift is an unbounded map. As a result, it is natural to study bilateral sequences of random objects, which are indexed by integers, rather than unilateral sequences. At the end of the paper, we will show Ryll-Nardzewski type theorems for monotone independence and boolean independence: Roughly speaking, an infinite bilateral sequence of random variables is monotonically (boolean) spreadable if and only if the variables are identically distributed and monotone (boolean) with respect to the conditional expectation onto its tail algebra. For an infinite sequence of noncommutative random variables, boolean spreadability is equivalent to boolean exchangeability.

Absence of algebraic relations and of zero divisors under the assumption of full non-microstates free entropy dimension

We show that in a tracial and finitely generated W⁎-probability space existence of conjugate variables excludes algebraic relations for the generators. Moreover, under the assumption of maximal non-microstates free entropy dimension, we prove that there are no zero divisors in the sense that the product of any non-commutative polynomial in the generators with any element from the von Neumann algebra is zero if and only if at least one of those factors is zero. In particular, this shows that in this case the distribution of any non-constant self-adjoint non-commutative polynomial in the generators does not have atoms.Questions on the absence of atoms for polynomials in non-commuting random variables (or for polynomials in random matrices) have been an open problem for quite a while. We solve this general problem by showing that maximality of free entropy dimension excludes atoms.