Order Freeness Research Articles

Finite free cumulants: Multiplicative convolutions, genus expansion and infinitesimal distributions

Given two polynomials p ( x ) , q ( x ) p(x), q(x) of degree d d , we give a combinatorial formula for the finite free cumulants of p ( x ) ⊠ d q ( x ) p(x)\boxtimes _d q(x) . We show that this formula admits a topological expansion in terms of non-crossing multi-annular permutations on surfaces of different genera. This topological expansion, on the one hand, deepens the connection between the theories of finite free probability and free probability, and in particular proves that ⊠ d \boxtimes _d converges to ⊠ \boxtimes as d d goes to infinity. On the other hand, borrowing tools from the theory of second order freeness, we use our expansion to study the infinitesimal distribution of certain families of polynomials which include Hermite and Laguerre, and draw some connections with the theory of infinitesimal distributions for real random matrices. Finally, building on our results we give a new short and conceptual proof of a recent result (see J. Hoskins and Z. Kabluchko [Exp. Math. (2021), pp. 1–27]; S. Steinerberger [Exp. Math. (2021), pp. 1–6]) that connects root distributions of polynomial derivatives with free fractional convolution powers.

A dynamical version of the SYK model and theq-Brownian motion

We extend recent results on the asymptotic eigenvalue distribution of the SYK model to the multivariate case and relate the limit of a dynamical version of the SYK model with the q-Brownian motion, a non-commutative deformation of classical Brownian motion. Furthermore, we extend the results for fluctuations to the multivariate setting and treat also higher correlation functions. The structure of our results for the sparse SYK random matrices resembles the formulas for higher order freeness for ordinary GUE random matrices.

Freeness over the diagonal and global fluctuations of complex Wigner matrices

We characterize the possible limiting 2nd order distributions of certain independent complex Wigner and deterministic matrices thanks to Voiculescu's notions of operator-valued freeness over the diagonal. If the Wigner matrices are Gaussian, Mingo and Speicher's notion of 2nd order freeness gives a universal rule, in terms of marginal 1st and 2nd order distribution. We adapt and reformulate this notion for operator-valued random variables in a 2nd order probability space. The Wigner matrices are assumed to be permutation invariant with null pseudo variance and the deterministic matrices to satisfy a restrictive property.

Modulus of continuity of controlled Loewner\u2013Kufarev equations and random matrices

First we introduce the two tau-functions which appeared either as the tau -function of the integrable hierarchy governing the Riemann mapping of Jordan curves or in conformal field theory and the universal Grassmannian. Then we discuss various aspects of their interrelation. Subsequently, we establish a novel connection between free probability, growth models and integrable systems, in particular for second order freeness, and summarise it in a dictionary. This extends the previous link between conformal maps and large N-matrix integrals to (higher) order free probability. Within this context of dynamically evolving contours, we determine a class of driving functions for controlled Loewner–Kufarev equations, which enables us to give a continuity estimate for the solution of such an equation when embedded into the Segal–Wilson Grassmannian.

A Characterization of High Order Freeness for Product Arrangements and Answers to Holm’s Questions

An m-free hyperplane arrangement is a generalization of a free arrangement. Holm asked the following two questions: (1)Does m-free imply (m + 1)-free for any arrangement? (2)Are all arrangements m-free for m large enough? In this paper, we characterize m-freeness for product arrangements, while we prove that all localizations of an m-free arrangement are m-free. From these results, we give answers to Holm’s questions.

Fluctuations of Marchenko–Pastur limit of random matrices with dependent entries

Pfaffel and Schlemm have investigated the sample covariance matrix with dependent entries motivated from moving average. Our former paper gave free probabilistic interpretation of it. In this paper, we suggest a new statistical hypothesis testing to coefficients of the moving average model using real second order freeness, which describes fluctuations of its almost sure limiting distribution.

Test of independence for high-dimensional random vectors based on freeness in block correlation matrices

In this paper, we are concerned with the independence test for $k$ high-dimensional sub-vectors of a normal vector, with fixed positive integer $k$. A natural high-dimensional extension of the classical sample correlation matrix, namely block correlation matrix, is proposed for this purpose. We then construct the so-called Schott type statistic as our test statistic, which turns out to be a particular linear spectral statistic of the block correlation matrix. Interestingly, the limiting behavior of the Schott type statistic can be figured out with the aid of the Free Probability Theory and the Random Matrix Theory. Specifically, we will bring the so-called real second order freeness for Haar distributed orthogonal matrices, derived in Mingo and Popa (2013)[10], into the framework of this high-dimensional testing problem. Our test does not require the sample size to be larger than the total or any partial sum of the dimensions of the $k$ sub-vectors. Simulated results show the effect of the Schott type statistic, in contrast to those statistics proposed in Jiang and Yang (2013)[8] and Jiang, Bai and Zheng (2013)[7], is satisfactory. Real data analysis is also used to illustrate our method.

Freeness and the transposes of unitarily invariant random matrices

We consider real second order freeness in the context of Haar unitary and unitarily invariant random matrices and their transposes. We show that a unitarily invariant ensemble with a second order limit distribution is asymptotically real second order free from the transposed ensemble. In particular we obtain the unexpected result that a unitarily invariant random matrix will be asymptotically free from its transpose.

Real second order freeness and Haar orthogonal matrices

We demonstrate the asymptotic real second order freeness of Haar distributed orthogonal matrices and an independent ensemble of random matrices. Our main result states that if we have two independent ensembles of random matrices with a real second order limit distribution and one of them is invariant under conjugation by an orthogonal matrix, then the two ensembles are asymptotically real second order free. This captures the known examples of asymptotic real second order freeness introduced by Redelmeier.

Rules of Designing Routing Metrics for Greedy, Face, and Combined Greedy-Face Routing

Different geographic routing protocols have different requirements on routing metric designs to ensure proper operation. Combining a wrong type of routing metrics with a geographic routing protocol may produce unexpected results, such as geographic routing loops and unreachable nodes. In this paper, we propose a novel routing algebra system to investigate the compatibilities between routing metrics and three geographic routing protocols including greedy, face, and combined greedy-face routing. Five important algebraic properties, respectively, named odd symmetry, transitivity, strict order, source independence, and local minimum freeness, are defined in this algebra system. Based on these algebraic properties, the necessary and sufficient conditions for loop-free, delivery-guaranteed, and consistent routing are derived when greedy, face, and combined greedy-face routing serve as packet forwarding schemes or as path discovery algorithms, respectively. Our work provides essential criteria for evaluating and designing geographic routing protocols.

Second order freeness and fluctuations of random matrices. III: Higher order freeness and free cumulants

We extend the relation between random matrices and free probability theory from the level of expectations to the level of all correlation functions (which are classical cumulants of traces of products of the matrices). We introduce the notion of “higher order freeness” and develop a theory of corresponding free cumulants. We show that two independent random matrix ensembles are free of arbitrary order if one of them is unitarily invariant. We prove \mathrm{R} -transform formulas for second order freeness. Much of the presented theory relies on a detailed study of the properties of “partitioned permutations”.

Second order freeness and fluctuations of random matrices: II. Unitary random matrices

We extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations. We show how the concept of “second order freeness”, which was introduced in Part I, allows one to understand global fluctuations of Haar distributed unitary random matrices. In particular, independence between the unitary ensemble and another ensemble goes in the large N limit over into asymptotic second order freeness. Two important consequences of our general theory are: (i) we obtain a natural generalization of a theorem of Diaconis and Shahshahani to the case of several independent unitary matrices; (ii) we can show that global fluctuations in unitarily invariant multi-matrix models are not universal.

Second order freeness and fluctuations of random matrices: I. Gaussian and Wishart matrices and cyclic Fock spaces

We extend the relation between random matrices and free probability theory from the level of expectations to the level of fluctuations. We introduce the concept of “second order freeness” and interpret the global fluctuations of Gaussian and Wishart random matrices by a general limit theorem for second order freeness. By introducing cyclic Fock space, we also give an operator algebraic model for the fluctuations of our random matrices in terms of the usual creation, annihilation, and preservation operators. We show that orthogonal families of Gaussian and Wishart random matrices are asymptotically free of second order.

Parsing Turkish using the lexical functional grammar formalism

This paper describes our work on parsing Turkish using thelexical-functional grammar formalism [11]. This work represents the first effort for wide-coverage syntactic parsing of Turkish. Our implementation is based on Tomita's parser developed at Carnegie Mellon University Center for Machine Translation. The grammar covers a substantial subset of Turkish including structurally simple and complex sentences, and deals with a reasonable amount of word order freeness. The complex agglutinative morphology of Turkish lexical structures is handled using a separate two-level morphological analyzer, which has been incorporated into the syntactic parser. After a discussion of the key relevant issues regarding Turkish grammar, we discuss aspects of our system and present results from our implementation. Our initial results suggest that our system can parse about 82% of the sentences directly and almost all the remaining with very minor pre-editing.