Noncommutative Probability Research Articles

Conditionally monotone cumulants via shuffle algebra

In this work, we study conditional monotone cumulants and additive convolution in the shuffle-algebraic approach to non-commutative probability. We describe c-monotone cumulants as an infinitesimal character and identify the c-monotone additive convolution as an associative operation in the set of pairs of characters in the dual of a double tensor Hopf algebra. In this algebraic framework, we understand previous results on c-monotone cumulants and prove a combinatorial formula that relates c-free and c-monotone cumulants. We also identify the notion of [Formula: see text]-Boolean cumulants in the shuffle-algebraic approach and introduce the corresponding notion of [Formula: see text]-monotone cumulants as a particular case of c-monotone cumulants.

Continuous asymmetric Doob inequalities in noncommutative symmetric spaces

Let (M,τ) be a noncommutative probability space equipped with a filtration (Mt)t∈[0,1] whose union is w⁎-dense in M, and let (Et)t∈[0,1] be the associated conditional expectations. We prove in the present paper that if the symmetric space E∈Int[Lp,Lq] with 1<p≤q<2 and E is 2(1−θ)-convex and w-concave with p<w<2, then the following holds:‖(Et(x))t∈[0,1]‖E(M;ℓ∞θ)≤CE,θ‖x‖HEc,x∈HEc(M) provided 1−p/2<θ<1. Similar result holds for x∈HEr(M). Moreover, if E∈Int[Lp,Lq] with 1<p≤q<2 and E is w-concave with 2<w<2p/(2−p), then for each x∈E(M) there exist y, z∈E(M) such that x=y+z and‖(Et(y))t∈[0,1]‖E(M;ℓ∞c)+‖(Et(z))t∈[0,1]‖E(M;ℓ∞r)≤cE‖x‖E(M). These results can be considered as continuous analogues of those due to Randrianantoanina et al. [33]. One of the key ingredients in our proof is a new decomposition theorem of E(M)-modules for general symmetric space E, which extends the known result of Junge and Sherman.

Two parameterized deformed Poisson type operator and the combinatorial moment formula

In this paper, we shall introduce two parameterized deformation of the classical Poisson random variable from the viewpoint of noncommutative probability, namely (q,s)-Poisson type operator (random variable) on the two parameterized deformed Fock space, namely, the (q,s)-Fock space constructed by the weighted q-deformation approach in [11,4] (see also [6]). The recurrence formula for the orthogonal polynomials of the (q,s)-deformed Poisson distribution is determined. Moreover we shall also give the combinatorial moment formula of the (q,s)-Poisson type operator by using the set partitions and the card arrangement technique with their statistics. Our method presented in this paper provides nice combinatorial interpretations to parameters, q and s. The deformation presented in this paper can be regarded as a generalization of the Al-Salam-Carlitz type, because the restricted case s=q recovers the q-Charlier polynomials of Al-Salam-Carlitz type appeared in combinatorics [17].

Limit distribution of partial transposition of block random matrices

It is well known that, under some assumptions, the limit distribution of block random matrices and their partial transposition converges to the distributions of random variables in some non-commutative probability spaces. Using free probability theory, we obtain the relation between the free cumulants of the corresponding random variables. As an application, we can derive a new family of co-completely positive and [Formula: see text]-positive maps by using the Wishart ensemble.

Non-commutative probability insights into the double-scaling limit SYK model with constant perturbations: moments, cumulants and q-independence

Extending the results of Wu (2022 J. Phys. A 55 415207), we study the double-scaling limit Sachdev–Ye–Kitaev model with an additional diagonal matrix with a fixed number c of nonzero constant entries θ. This constant diagonal term can be rewritten in terms of Majorana fermion products. Its specific formula depends on the value of c. We find exact expressions for the moments of this model. More importantly, by proposing a moment-cumulant relation, we reinterpret the effect of introducing a constant term in the context of non-commutative probability theory. This gives rise to a q~ dependent mixture of independences within the moment formula. The parameter q~ , derived from the q-Ornstein–Uhlenbeck (q-OU) process, controls this transformation. It interpolates between classical independence ( q~=1 ) and Boolean independence ( q~=0 ). The underlying combinatorial structures of this model provide the non-commutative probability connections. Additionally, we explore the potential relation between these connections and their gravitational path integral counterparts.

Algebraic groups in non-commutative probability theory revisited

Algebraic groups in non-commutative probability theory revisited

Some Limiting Laws in Non-Commutative Probability

In this article, we provide some new limiting laws related to the free multiplicative law of large numbers and involving free and Boolean additive convolutions. Some examples of these limiting laws are presented within the framework of non-commutative probability theory.

Generating series and matrix models for meandric systems with one shallow side

In this article, we investigate meandric systems having one shallow side: the arch configuration on that side has depth at most two. This class of meandric systems was introduced and extensively examined by I. P. Goulden, A. Nica, and D. Puder [Int. Math. Res. Not. IMRN 2020 (2020), 983–1034]. Shallow arch configurations are in bijection with the set of interval partitions. We study meandric systems by using moment-cumulant transforms for non-crossing and interval partitions, corresponding to the notions of free and Boolean independence, respectively, in non-commutative probability. We obtain formulas for the generating series of different classes of meandric systems with one shallow side by explicitly enumerating the simpler, irreducible objects. In addition, we propose random matrix models for the corresponding meandric polynomials, which can be described in the language of quantum information theory, in particular that of quantum channels.

Navigating the realm of noncommutative probability: Historical perspectives and future directionsstorical perspectives and future directions

This review article provides a comprehensive overview of the fascinating field of noncommutative probability theory, tracing its evolution from its inception in the early 1980s by Romanian-American mathematician Dan Voiculescu to its current state of prominence in mathematics. Through a meticulous examination of seminal works and recent advancements, we explore the key concepts, methodologies, and significant developments in this field, emphasizing the combinatorial aspects of noncommutative probability spaces, including non-crossing partitions and linked partitions. This exploration encompasses various aspects, including analytical methods, operator algebras, random matrices, and combinatorial structures. Additionally, it concludes with the current understanding and potential directions for future research.

Cumulants, spreadability and the Campbell–Baker–Hausdorff series

We define spreadability systems as a generalization of exchangeability systems in order to unify various notions of independence and cumulants known in noncommutative probability. In particular, our theory covers monotone independence and monotone cumulants which do not satisfy exchangeability. To this end we study generalized zeta and Möbius functions in the context of the incidence algebra of the semilattice of ordered set partitions and prove an appropriate variant of Faà di Bruno’s theorem. With the aid of this machinery we show that our cumulants cover most of the previously known cumulants. Due to noncommutativity of independence the behaviour of these cumulants with respect to independent random variables is more complicated than in the exchangeable case and the appearance of Goldberg coefficients exhibits the role of the Campbell–Baker–Hausdorff series in this context. Moreover, we exhibit an interpretation of the Campbell–Baker–Hausdorff series as a sum of cumulants in a particular spreadability system, thus providing a new derivation of the Goldberg coefficients.

Continuous evolution families

Recently in relation to the theory of non-commutative probability, the notion of evolution family { ω s , t } s ≤ t \{\omega _{s,t}\}_{s \le t} is generalized assuming only continuity in parameters, namely ( s , t ) ↦ ω s , t (s,t) \mapsto \omega _{s,t} is continuous with respect to locally uniform convergence on a planar domain. In this article we present various equivalence conditions to the continuous evolution families concerned with the left and right parameters. We also provide an example of a discontinuous evolution family in the last section.

Shifted Substitution in Non-Commutative Multivariate Power Series with a View Toward Free Probability

We study a particular group law on formal power series in non-commuting variables induced by their interpretation as linear forms on a suitable graded connected word Hopf algebra. This group law is left-linear and is therefore associated to a pre-Lie structure on formal power series. We study these structures and show how they can be used to recast in a group theoretic form various identities and transformations on formal power series that have been central in the context of non-commutative probability theory, in particular in Voiculescu's theory of free probability.

Fertilitopes

We introduce tools from discrete convexity theory and polyhedral geometry into the theory of West’s stack-sorting map s. Associated to each permutation pi is a particular set mathcal V(pi ) of integer compositions that appears in a formula for the fertility of pi , which is defined to be |s^{-1}(pi )|. These compositions also feature prominently in more general formulas involving families of colored binary plane trees called troupes and in a formula that converts from free to classical cumulants in noncommutative probability theory. We show that mathcal V(pi ) is a transversal discrete polymatroid when it is nonempty. We define the fertilitope of pi to be the convex hull of mathcal V(pi ), and we prove a surprisingly simple characterization of fertilitopes as nestohedra arising from full binary plane trees. Using known facts about nestohedra, we provide a procedure for describing the structure of the fertilitope of pi directly from pi using Bousquet-Mélou’s notion of the canonical tree of pi . As a byproduct, we obtain a new combinatorial cumulant conversion formula in terms of generalizations of canonical trees that we call quasicanonical trees. We also apply our results on fertilitopes to study combinatorial properties of the stack-sorting map. In particular, we show that the set of fertility numbers has density 1, and we determine all infertility numbers of size at most 126. Finally, we reformulate the conjecture that sum _{sigma in s^{-1}(pi )}x^{textrm{des}(sigma )+1} is always real-rooted in terms of nestohedra, and we propose natural ways in which this new version of the conjecture could be extended.

Shuffle Algebras and Non-Commutative Probability for Pairs of Faces

One can build an operatorial model for freeness by considering either the right-handed or the left-handed representation of algebras of operators acting on the free product of the underlying pointed Hilbert spaces. Considering both at the same time, that is, computing distributions of operators in the algebra generated by the left- and right-handed representations, led Voiculescu in 2013 to define and study bifreeness and, in the sequel, triggered the development of an extension of noncommutative probability now frequently referred to as multi-faced (two-faced in the example given above). Many examples of two-faced independences emerged these past years. Of great interest to us are biBoolean, bifree and type I bimonotone independences. In this paper, we extend the preLie calculus pertaining to free, Boolean, and monotone moment-cumulant relations initiated by K. Ebrahimi-Fard and F. Patras to their above-mentioned two-faced equivalents.

Multiplicative and semi-multiplicative functions on non-crossing partitions, and relations to cumulants

We consider the group (G,⁎) of unitized multiplicative functions in the incidence algebra of non-crossing partitions, where “⁎” denotes the convolution operation. We introduce a larger group (G˜,⁎) of unitized functions from the same incidence algebra, which satisfy a weaker semi-multiplicativity condition. The natural action of G˜ on sequences of multilinear functionals of a non-commutative probability space captures the combinatorics of transitions between moments and some brands of cumulants that are studied in the non-commutative probability literature. We use the framework of G˜ in order to explain why the multiplication of free random variables can be very nicely described in terms of Boolean cumulants and more generally in terms of t-Boolean cumulants, a one-parameter interpolation between free and Boolean cumulants arising from work of Bożejko and Wysoczanski.It is known that the group G can be naturally identified as the group of characters of the Hopf algebra Sym of symmetric functions. We show that G˜ can also be identified as group of characters of a Hopf algebra T, which is an incidence Hopf algebra in the sense of Schmitt. Moreover, the inclusion G⊆G˜ turns out to be the dual of a natural bialgebra homomorphism from T onto Sym.

Fock space associated with quadrabasic Hermite orthogonal polynomials

This paper introduces a new idea for constructing operators associated with a certain class of probability measures. Special cases include known classical ones and noncommutative probability. The main example, derived from Feller’s book, is the hyperbolic

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.

On the Signature of a Path in an Operator Algebra

We introduce a class of operators associated with the signature of a smooth path $X$ with values in a $C^{\star}$ algebra $\mathcal{A}$. These operators serve as the basis of Taylor expansions of solutions to controlled differential equations of interest in noncommutative probability. They are defined by fully contracting iterated integrals of $X$, seen as tensors, with the product of $\mathcal{A}$. Were it considered that partial contractions should be included, we explain how these operators yield a trajectory on a group of representations of a combinatorial Hopf monoid. To clarify the role of partial contractions, we build an alternative group-valued trajectory whose increments embody full-contractions operators alone. We obtain therefore a notion of signature, which seems more appropriate for noncommutative probability.

Convergence for noncommutative rational functions evaluated in random matrices

One of the main applications of free probability is to show that for appropriately chosen independent copies of $d$ random matrix models, any noncommutative polynomial in these $d$ variables has a spectral distribution that converges asymptotically and can be described with the help of free probability. This paper aims to show that this can be extended to noncommutative rational functions, answering an open question by Roland Speicher. This paper also provides a noncommutative probability approach to approximating the free field. At the algebraic level, its construction relies on the approximation by generic matrices. On the other hand, it admits many embeddings in the algebra of operators affiliated with a $II_1$ factor. A consequence of our result is that, as soon as the generators admit a random matrix model, the approximation of any self-adjoint noncommutative rational function by generic matrices can be upgraded at the level of convergence in distribution.

Motzkin path decompositions of functionals in noncommutative probability

We study the decomposition of free random variables in terms of their orthogonal replicas from a new perspective. First, we show that the mixed moments of orthogonal replicas with respect to the normalized linear functional [Formula: see text] are naturally described in terms of Motzkin paths identified with reduced Motzkin words [Formula: see text]. Using this fact, we demonstrate that the mixed moments of order [Formula: see text] of free random variables with respect to the free product of normalized linear functionals are sums of the mixed moments of the orthogonal replicas of these variables with respect to [Formula: see text] with summation extending over [Formula: see text], the set of reduced Motzkin paths of length [Formula: see text]. One of the applications of this formula is a decomposition formula for mixed moments of free random variables in terms of their boolean cumulants which corresponds to the decomposition of the lattice [Formula: see text] into sublattices [Formula: see text] of partitions which are monotonically adapted to colors in [Formula: see text]. The linear functionals defined by the mixed moments of orthogonal replicas and indexed by reduced Motzkin words play the role of a generating set of the space of product functionals in which the boolean product corresponds to constant Motzkin paths and the free product corresponds to all Motzkin paths.