Bernadette Krawczyk, Roland Speicher
We derive a formula for expressing free cumulants whose entries are products of random variables in terms of the lattice structure of non-crossing partitions. We show the usefulness of that result by giving direct and conceptually simple proofs for a lot of results about $R$-diagonal elements. Our investigations do not assume the trace property for the considered linear functionals.
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The Quantum Symmetric Simple Exclusion Process (QSSEP) is a model of quantum particles hopping on a finite interval and satisfying the exclusion principle. Recently Bernard and Jin have studied the fluctuations of the invariant measure for this process, when the number of sites goes to infinity. These fluctuations are encoded into polynomials, for which they have given equations and proved that these equations determine the polynomials completely. In this paper, I give an explicit combinatorial formula for these polynomials, in terms of Schröder trees. I also show that, quite surprisingly, the…
We express classical, free, Boolean and monotone cumulants in terms of each other, using combinatorics of heaps, pyramids, Tutte polynomials and permutations. We completely determine the coefficients of these formulas with the exception of the formula for classical cumulants in terms of monotone cumulants whose coefficients are only partially computed.
Free cumulants were introduced as the proper analog of classical cumulants in the theory of free probability. There is a mix of similarities and differences, when one considers the two families of cumulants. Whereas the combinatorics of classical cumulants is well expressed in terms of set partitions, the one of free cumulants is described, and often introduced in terms of non-crossing set partitions. The formal series approach to classical and free cumulants also largely differ. It is the purpose of the present article to put forward a different approach to these phenomena. Namely, we show th…
Unlike classical and free independence, the boolean and monotone notions of independence lack of the property of independent constants. In the scalar case, this leads to restrictions for the central limit theorems, as observed by F. Oravecz. We characterize the property of independent constants from a combinatorial point of view, based on cumulants and set partitions. This characterization also holds for the operator-valued extension. Our considerations lead rather directly to very mild variations of boolean and monotone cumulants, where constants are now independent. These alternative probabi…
Construction of product $*$-probability spaces via free cumulants
本文以混合自由累积量消失作为自由性定义，利用自由累积量而非矩来构造 *-概率空间的自由乘积。
Decomposition of free cumulants
引入适应Motzkin路径的非交叉分拆格，定义算子值Motzkin累积量，通过Möbius反演公式细化自由累积量与布尔累积量的关系，并应用于自由概率得到自由累积量的加性分解。
We study joint free cumulants of u_t and u_t^{*}, where u_t is a free unitary Brownian motion at time t. We determine explicitly some special families of such cumulants. On the other hand, for a general joint cumulant of u_t and u_t^{*}, we "calculate the derivative" for t going to infinity, when u_t approaches a Haar unitary. In connection to the latter calculation we put into evidence an "infinitesimal determining sequence" which naturally accompanies an arbitrary R-diagonal element in a tracial *-probability space.
We study how Boolean cumulants can be used in order to address operations with freely independent random variables, particularly in connection to the $*$-distribution of the product of two selfadjoint freely independent random variables, and in connection to the distribution of the anticommutator of such random variables.
We establish connections between the lattices of non-crossing partitions of type B introduced by V. Reiner, and the framework of the free probability theory of D. Voiculescu. Lattices of non-crossing partitions (of type A, up to now) have played an important role in the combinatorics of free probability, primarily via the non-crossing cumulants of R. Speicher. Here we introduce the concept of {\em non-crossing cumulant of type B;} the inspiration for its definition is found by looking at an operation of ``restricted convolution of multiplicative functions'', studied in parallel for functions o…
In this paper we define cumulants for finite free convolution. We give a moment-cumulant formula and show that these cumulants satisfy desired properties: they are additive with respect to finite free convolution and they approach free cumulants as the dimension goes to infinity.
The q-semicircular distribution is a probability law that interpolates between the Gaussian law and the semicircular law. There is a combinatorial interpretation of its moments in terms of matchings where q follows the number of crossings, whereas for the free cumulants one has to restrict the enumeration to connected matchings. The purpose of this article is to describe combinatorial properties of the classical cumulants. We show that like the free cumulants, they are obtained by an enumeration of connected matchings, the weight being now an evaluation of the Tutte polynomial of a so-called c…
Cumulant-cumulant relations in free probability theory from Magnus' expansion
本文利用 pre-Lie 代数上的 Magnus 展开，结合连续 Baker-Campbell-Hausdorff 公式，给出了用自由和布尔累积量闭式表达多元单调累积量的方法，并引入了非交叉划分上的新构造与统计量。
Boolean, free and monotone cumulants as well as relations among them, have proven to be important in the study of non-commutative probability theory. Quite notably, Boolean cumulants were successfully used to study free infinite divisibility via the Boolean Bercovici--Pata bijection. On the other hand, in recent years the concept of infinitesimal non-commutative probability has been developed, together with the notion of infinitesimal cumulants which can be useful in the context of combinatorial questions. In this paper, we show that the known relations among free, Boolean and monotone cumulan…
Motivated by recent work on mixtures of classical and free probabilities, we introduce and study the notion of $ε$-noncrossing partitions. It is shown that the set of such partitions forms a lattice, which interpolates as a poset between the poset of partitions and the one of noncrossing partitions. Moreover, $ε$-cumulants are introduced and shown to characterize the notion of $ε$-independence.
We provide an unifying polynomial expression giving moments in terms of cumulants, and viceversa, holding in the classical, boolean and free setting. This is done by using a symbolic treatment of Abel polynomials. As a by-product, we show that in the free cumulant theory the volume polynomial of Pitman and Stanley plays the role of the complete Bell exponential polynomial in the classical theory. Moreover via generalized Abel polynomials we construct a new class of cumulants, including the classical, boolean and free ones, and the convolutions linearized by them. Finally, via an umbral Fourier…
Convolution, cumulants and infinitesimal generators in the formal power series ring
本文探讨了形式幂级数中t变形卷积和累积量的概念，推导了大数定律和中心极限定理的t变形版本，并建立了与经典、自由和有限自由概率理论的结构相似性。
The theory of cumulants is revisited in the "Rota way", that is, by following a combinatorial Hopf algebra approach. Monotone, free, and boolean cumulants are considered as infinitesimal characters over a particular combinatorial Hopf algebra. The latter is neither commutative nor cocommutative, and has an underlying unshuffle bialgebra structure which gives rise to a shuffle product on its graded dual. The moment-cumulant relations are encoded in terms of shuffle and half-shuffle exponentials. It is then shown how to express concisely monotone, free, and boolean cumulants in terms of each oth…
Finite Free Cumulants: Multiplicative Convolutions, Genus Expansion and Infinitesimal Distributions
本文给出了多项式乘积卷积的有限自由累积量的组合公式及其拓扑展开，揭示了有限自由概率与自由概率的联系，并利用该展开研究了多项式的无穷小分布，最后为多项式导数根分布与自由分数卷积幂的联系提供了新证明。
We study the coefficients in the expansion of Jack polynomials in terms of power sums. We express them as polynomials in the free cumulants of the transition measure of an anisotropic Young diagram. We conjecture that such polynomials have nonnegative integer coefficients. This extends recent results about normalized characters of the symmetric group.
Free cumulants were introduced by Speicher as a proper analog of classical cumulants in Voiculescu's theory of free probability. The relation between free moments and free cumulants is usually described in terms of Moebius calculus over the lattice of non-crossing partitions. In this work we explore another approach to free cumulants and to their combinatorial study using a combinatorial Hopf algebra structure on the linear span of non-crossing partitions. The generating series of free moments is seen as a character on this Hopf algebra. It is characterized by solving a linear fixed point equa…
Cumulants in rectangular finite free probability and beta-deformed singular values
本文定义了(n,d)-矩形矩量，证明了矩量-矩公式，并展示其在有限自由概率中的线性化作用及在特定参数下收敛于自由概率中的q-矩形自由矩量，应用于对称经验根分布极限的研究。
A combinatorial formula is derived which expresses free cumulants in terms of classical comulants. As a corollary, we give a combinatorial interpretation of free cumulants of classical distributions, notably Gaussian and Poisson distributions. The latter count connected pairings and connected set partitions respectively. The proof relies on Moebius inversion on the partition lattice.
The natural forms of the Leibniz rule for the $k$th derivative of a product and of Faà di Bruno's formula for the $k$th derivative of a composition involve the differential operator $\partial^k/\partial x_1 ... \partial x_k$ rather than $d^k/dx^k$, with no assumptions about whether the variables $x_1,...,x_k$ are all distinct, or all identical, or partitioned into several distinguishable classes of indistinguishable variables. Coefficients appearing in forms of these identities in which some variables are indistinguishable are just multiplicities of indistinguishable terms (in particular, if a…
We count the n-strand braids whose normal decomposition has length at most two in the dual braid monoid B_n+* by reducing the question to a computation of free cumulants for a product of independent variables, for which we establish a general formula.
Combinatorics and loop equations for antisymmetrised and Hermitised matrix product ensembles
本研究针对反对称化与厄米化矩阵乘积系综，构造由混合累积量计数的带状图和星座图，并在m=1时给出生成函数的圈方程刻画。
Cumulants linearize convolution of measures. We use a formula of Good to define noncommutative cumulants in a very general setting.It turns out that the essential property needed is exchangeability of random variables. Roughly speaking the formula says that cumulants are moments of a certain ``discrete Fourier transform'' of a random variable. This provides a simple unified method to understand the known examples of cumulants, like classical, free cumulants and various q-cumulants.
Functional relations for higher-order free cumulants
本文扩展自由概率理论，利用涉及双单调 Hurwitz 数的主变换与半无限楔形技术，建立高阶自由累积量生成级数与矩之间的函数关系，并在带尖峰 GUE 系综上验证。
The functional equation defining the free cumulants in free probability is lifted successively to the noncommutative Faà di Bruno algebra, and then to the group of a free operad over Schröder trees. This leads to new combinatorial expressions, which remain valid for operator-valued free probability. Specializations of these expressions give back Speicher's formula in terms of noncrossing partitions, and its interpretation in terms of characters due to Ebrahimi-Fard and Patras.
In this paper we collect a few results about exchangeability systems in which crossing cumulants vanish, which we call noncrossing exchangeability systems. The main result is a free version of De Finetti's theorem, characterising amalgamated free products as noncrossing exchangeability systems which satisfy a so-called weak singleton condition. The main tool in the proof is an $L^p$-inequality with uniformly bounded constants for i.i.d. sequences in noncrossing exchangeability systems.
Boolean, Free, and Classical Cumulants as Tree Enumerations
本文通过 weighted troupes 将 Defant 的二叉平面树枚举关系从单变量推广到多变量 cumulants，并统一刻画 Boolean cumulants，回答了 troupe transform 相关问题。
Higher order free moments and cumulants, introduced by Collins, Mingo, Śniady and Speicher in 2006, describe the fluctuations of unitarily invariant random matrices in the limit of infinite size. The functional relations between their generating functions were only found last year by Borot, Garcia-Failde, Charbonnier, Leid and Shadrin and a combinatorial derivation is still missing. We simplify these relations and show how their combinatorial derivation reduces to the computation of generating functions of planar non-separable hypermaps with prescribed vertex valencies and weighted hyper-edges…
Forest formulas that generalize Zimmermann's forest formula in quantum field theory have been obtained for the computation of the antipode in the dual of enveloping algebras of pre-Lie algebras. In this work, largely motivated by Murua's analysis of the Baker-Campbell-Hausdorff formula, we show that the same ideas and techniques generalize and provide effective tools to handle computations in these algebras, which are of utmost importance in numerical analysis and related areas. We illustrate our results by studying the action of the pre-Lie exponential and the Magnus operator in the free pre-…
Free cumulants and freeness for unitarily invariant random tensors
本文研究局部酉不变随机张量的渐近描述，通过矩-累积量公式定义了纯张量与混合张量的张量自由累积量，证明其可加性，并基于此提出张量自由性概念，推广了非交换概率空间。
This paper introduces a simple and computationally efficient algorithm for conversion formulae between moments and cumulants. The algorithm provides just one formula for classical, boolean and free cumulants. This is realized by using a suitable polynomial representation of Abel polynomials. The algorithm relies on the classical umbral calculus, a symbolic language introduced by Rota and Taylor in 1994, that is particularly suited to be implemented by using software for symbolic computations. Here we give a MAPLE procedure. Comparisons with existing procedures, especially for conversions betwe…
Second order free cumulants: product, commutator, and anti-commutator
本文研究两个二阶自由独立随机变量的乘积、交换子与反对易子，给出其二阶自由累积量的一般求和公式，并应用于二阶自由半圆变量。
Free cumulants are nice and useful functionals of the shape of a Young diagram, in particular they give the asymptotics of normalized characters of symmetric groups S(n) in the limit n\to\infty. We give an explicit combinatorial formula for normalized characters of the symmetric groups in terms of free cumulants. We also express characters in terms of Frobenius coordinates. Our formulas involve counting certain factorizations of a given permutation. The main tool are Stanley polynomials which give values of characters on multirectangular Young diagrams. Résumé. Les cumulants libres sont des fo…
Finite $N$ precursors of the free cumulants
本文研究了 $N\times N$ 矩阵空间上的 $\mathrm{U}(N)$ 不变多项式，作为自由累积量的有限 $N$ 前体，证明了其加性及高斯权重下的 Wick 规则等性质，并基于 HCIZ 积分计算了描述共轭轨道加法行为的余积。
We study zonal characters which are defined as suitably normalized coefficients in the expansion of zonal polynomials in terms of power-sum symmetric functions. We show that the zonal characters, just like the characters of the symmetric groups, admit a nice combinatorial description in terms of Stanley's multirectangular coordinates of Young diagrams. We also study the analogue of Kerov polynomials, namely we express the zonal characters as polynomials in free cumulants and we give an explicit combinatorial interpretation of their coefficients. In this way, we prove two recent conjectures of …
Finite free convolution via reproducing kernels and squarefree algebras
本文通过多项式空间上的再生核内积和无平方代数上的多线性模型，给出了Marcus、Spielman和Srivastava的有限自由卷积的结构性解释，并引入了一类由高阶累积量非正性刻画的多线性多项式。
A formula expressing free cumulants in terms of the Jacobi parameters of the corresponding orthogonal polynomials is derived. It combines Flajolet's theory of continued fractions and Lagrange inversion. For the converse we discuss Gessel-Viennot theory to express Hankel determinants in terms of various cumulants.
We give a direct and simple proof of Touchard's continued fraction, provide an extension of it, and transform it into similar expansions related to Motzkin and Schroeder numbers. Another proof is then given that uses only induction. We use this machinery on two examples that appear in recent papers of Josuat-Verges; with an additional parameter, these two can be treated simultaneously.
The Δ-convolution of real probability measures, introduced by Bożejko, generalizes both free and boolean convolutions. It is linearized by the Δ-cumulants, and Yoshida gave a combinatorial formula for moments in terms of Δ-cumulants, that implicitly defines the latter. It relies on the definition of an appropriate weight on noncrossing partitions. We give here two different expressions for the Δ-cumulants: the first one is a simple variant of Lagrange inversion formula, and the second one is a combinatorial inversion of Yoshida's formula involving Schröder trees.
We study the combinatorics of pseudoline arrangements in the real projective plane. Our focus lies on two classes of arrangements: simplicial arrangements and arrangements whose characteristic polynomials have only real roots. We derive inequalities involving the $t$-vectors of the arrangements in consideration. As application, we obtain some finiteness and classification results. Moreover, we are able to prove the Dirac Motzkin Conjecture for real pseudoline arrangements whose charateristic polynomials split over $\mathbb{R}$.
In this paper, we construct a bijection from a set of bounded free Motzkin paths to a set of bounded Motzkin prefixes that induces a bijection from a set of bounded free Dyck paths to a set of bounded Dyck prefixes. We also give bijections between a set of bounded cornerless Motzkin paths and a set of $t$-core partitions, and a set of bounded cornerless symmetric Motzkin paths and a set of self-conjugate $t$-core partitions. As an application, we get explicit formulas for the number of ordinary and self-conjugate $t$-core partitions with a fixed number of corners.
We study the generalization of shifted Jack polynomials to arbitrary multiplicity free spaces. In a previous paper (math.RT/0006004) we showed that these polynomials are eigenfunctions for commuting difference operators. Our central result now is the "transposition formula", a generalization of Okounkov's binomial theorem (q-alg/9608021) for shifted Jack polynomials. From this formula, we derive an interpolation formula, an evaluation formula, a scalar product, a binomial theorem, and properties of the algebra generated by the multiplication and difference operators.
This paper is devoted to the computations of some relevant quantities associated with the free unitary Brownian motion. Using the Lagrange inversion formula, we first derive an explicit expression for its alternating star cumulants of even lengths and relate them to those having odd lengths by means of a summation formula for the free cumulants with product as entries. Next, we use again this formula together with a generating series for Laguerre polynomials in order to compute the Taylor coefficients of the reciprocal of the $R$-transform of the free Jacobi process associated with a single pr…
Free Meixner states are a class of functionals on non-commutative polynomials introduced in math.CO/0410482. They are characterized by a resolvent-type form for the generating function of their orthogonal polynomials, by a recursion relation for those polynomials, or by a second-order non-commutative differential equation satisfied by their free cumulant functional. In this paper, we construct an operator model for free Meixner states. By combinatorial methods, we also derive an operator model for their free cumulant functionals. This, in turn, allows us to construct a number of examples. Many…
In the theory of hyperplane arrangements, the most important and difficult problem is the combinatorial dependency of several properties. In this atricle, we prove that Terao's celebrated addition-deletion theorem for free arrangements is combinatorial, i.e., whether you can apply it depends only on the intersection lattice of arrangements. The proof is based on a classical technique. Since some parts are already completed recently, we prove the rest part, i.e., the combinatoriality of the addition theorem. As a corollary, we can define a new class of free arrangements called the additionally …
Several sequences of free cumulants that count binary plane trees correspond to sequences of classical cumulants that count the decreasing versions of the same trees. Using two new operations on colored binary plane trees that we call insertion and decomposition, we prove that this surprising phenomenon holds for families of trees that we call troupes. We give a simple characterization of troupes, which provide a broad framework for generalizing several of the results known about West's stack-sorting map $s$. Indeed, we give new proofs of some of the main techniques that have been developed fo…
We study the free (associative, non-commutative) Baxter algebra on one generator. The first explicit description of this object is due to Ebrahimi-Fard and Guo. We provide an alternative description in terms of a certain class of trees, which form a linear basis for this algebra. We use this to treat other related cases, particularly that in which the Baxter map is required to be quasi-idempotent, in a unified manner. Each case corresponds to a different class of trees. Our main focus is on the underlying combinatorics. In several cases, we provide bijections between our various classes of tre…
We show that the deletion theorem of a free arrangement is combinatorial, i.e., whether we can delete a hyperplane from a free arrangement keeping freeness depends only on the intersection lattice. In fact, we give an explicit sufficient and necessary condition for the deletion theorem in terms of characteristic polynomials. This gives a lot of corollaries including the existence of free filtrations. The proof is based on the result about the form of minimal generators of a logarithmic derivation module of a multiarrangement which satisfies the $b_2$-equality.
Kerov polynomials describe normalized irreducible characters of the symmetric groups in terms of the free cumulants associated with Young diagrams. We suggest well-suited counterparts of the Kerov polynomials in spin (or projective) representation settings. We show that spin analogues of irreducible characters are polynomials in even free cumulants associated with double diagrams of strict partitions. Moreover, we present a conjecture for the positivity of their coefficients.
Snakes are analogues of alternating permutations defined for any Coxeter group. We study these objects from the point of view of combinatorial Hopf algebras, such as noncommutative symmetric functions and their generalizations. The main purpose is to show that several properties of the generating functions of snakes, such as differential equations or closed form as trigonometric functions, can be lifted at the level of noncommutative symmetric functions or free quasi-symmetric functions. The results take the form of algebraic identities for type B noncommutative symmetric functions, noncommuta…
We study the asymptotic behavior of the free cumulants (in the sense of free probability theory of Voiculescu) of Jucys--Murphy elements--or equivalently--of the transition measure associated with a Young diagram. We express these cumulants in terms of normalized characters of the appropriate representation of the symmetric group S_q. Our analysis considers the case when the Young diagrams rescaled by q^{-1/2} converge towards some prescribed shape. We find explicitly the second order asymptotic expansion and outline the algorithm which allows to find the asymptotic expansion of any order. As …
The probabilistic method is a technique for proving combinatorial existence results by means of showing that a randomly chosen object has the desired properties with positive probability. A particularly powerful probabilistic tool is the Lovász Local Lemma (the LLL for short), which was introduced by Erdős and Lovász in the mid-1970s. Here we develop a version of the LLL that can be used to prove the existence of continuous colorings. We then give several applications in Borel and topological dynamics. * Seward and Tucker-Drob showed that every free Borel action $Γ\curvearrowright X$ of a coun…
In this work, some classical results of the pfaffian theory of the dimer model based on the work of Kasteleyn, Fisher and Temperley are introduced in a fermionic framework. Then we shall detail the bosonic formulation of the model {\it via} the so-called height mapping and the nature of boundary conditions is unravelled. The complete and detailed fermionic solution of the dimer model on the square lattice with an arbitrary number of monomers is presented, and finite size effect analysis is performed to study surface and corner effects, leading to the extrapolation of the central charge of the …
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