$${\fancyscript{R}}$$ -diagonal dilation semigroups

We show that if A is a Hilbert–space operator, then the set of all projections onto hyperinvariant subspaces of A, which is contained in the von Neumann algebra υN(A) that is generated by A, is independent of the representation of υ N(A), thought of as an abstract W*–algebra. We modify a technique of Foias, Ko, Jung and Pearcy to get a method for finding nontrivial hyperinvariant subspaces of certain operators in finite von Neumann algebras. We introduce the B–circular operators as a special case of Speicher's B–Gaussian operators in free probability theory, and we prove several results about a B–circular operator z, including formulas for the B–valued Cauchy– and R–transforms of z*z. We show that a large class of L∞([0,1])–circular operators in finite von Neumann algebras have nontrivial hyperinvariant subspaces, and that another large class of them can be embedded in the free group factor L(F3). These results generalize some of what is known about the quasinilpotent DT–operator.

Unitary designs have become a vital tool for investigating pseudorandomness, since they approximate the statistics of the uniform Haar ensemble. Despite their central role in quantum information, their relations to quantum chaotic evolution and, in particular, to the eigenstate thermalization hypothesis (ETH) are still largely debated issues. This work provides a bridge between the latter and k designs through free probability theory. First, by introducing the more general notion of k-freeness, we show that it can be used as an alternative probe to designs. In turn, free probability theory comes with several tools, useful, for instance, for the calculation of mixed moments or the so-called k-fold quantum channels. Our second result is the connection to quantum dynamics. Quantum ergodicity and, correspondingly, ETH apply to a restricted class of physical observables, as already discussed in the literature. In this spirit, we show that unitary evolution with generic Hamiltonians always leads to freeness at sufficiently long times but only when the operators considered are restricted within the ETH class. Our results provide a direct link between unitary designs, quantum chaos, and the eigenstate thermalization hypothesis and shed new light on the universality of late-time quantum dynamics.

We present a model unifying general relativity and quantum mechanics. The model is based on the (noncommutative) algebra \mbox{\cal A} on the groupoid \Gamma = E \times G where E is the total space of the frame bundle over spacetime, and G the Lorentz group. The differential geometry, based on derivations of \mbox{\cal A}, is constructed. The eigenvalue equation for the Einstein operator plays the role of the generalized Einstein's equation. The algebra \mbox{\cal A}, when suitably represented in a bundle of Hilbert spaces, is a von Neumann algebra \mathcal{M} of random operators representing the quantum sector of the model. The Tomita-Takesaki theorem allows us to define the dynamics of random operators which depends on the state \phi . The same state defines the noncommutative probability measure (in the sense of Voiculescu's free probability theory). Moreover, the state \phi satisfies the Kubo-Martin-Schwinger (KMS) condition, and can be interpreted as describing a generalized equilibrium state. By suitably averaging elements of the algebra \mbox{\cal A}, one recovers the standard geometry of spacetime. We show that any act of measurement, performed at a given spacetime point, makes the model to collapse to the standard quantum mechanics (on the group G). As an example we compute the noncommutative version of the closed Friedman world model. Generalized eigenvalues of the Einstein operator produce the correct components of the energy-momentum tensor. Dynamics of random operators does not ``feel'' singularities.

Commutative shuffle products are known to be intimately related to universal formulas for products, exponentials and logarithms in group theory as well as in the theory of free Lie algebras, such as, for instance, the Baker-Campbell-Hausdorff formula or the analytic expression of a Lie group law in exponential coordinates in the neighbourhood of the identity. Non-commutative shuffle products happen to have similar properties with respect to pre-Lie algebras. However, the situation is more complex since in the non-commutative framework three exponential-type maps and corresponding logarithms are naturally defined. This results in several new formal group laws together with new operations - for example a new notion of adjoint action particularly well fitted to the new theory. These developments are largely motivated by various constructions in non-commutative probability theory. The second part of the article is devoted to exploring and deepening this perspective. We illustrate our approach by revisiting universal products from a group-theoretical viewpoint, including additive convolution in monotone, free and boolean probability, as well as the Bercovici-Pata bijection and subordination products.

We provide a unified and self-contained treatment of several of the recent uniqueness theorems for the group measure space decomposition of a II1 factor. We single out a large class of groups Γ, characterized by a one-cohomology property, and prove that for every free ergodic probability measure preserving action of Γ the associated II1 factor has a unique group measure space Cartan subalgebra up to unitary conjugacy. Our methods follow closely a recent article of Chifan–Peterson, but we replace the usage of Peterson’s unbounded derivations by Thomas Sinclair’s dilation into a malleable deformation by a one-parameter group of automorphisms.

We prove that for any free ergodic probability measure-preserving action \({\mathbb{F}_n \curvearrowright (X, \mu)}\) of a free group on n generators \({\mathbb{F}_n, 2\leq n \leq \infty}\), the associated group measure space II1 factor \({L^\infty (X)\rtimes \mathbb{F}_n}\) has L∞(X) as its unique Cartan subalgebra, up to unitary conjugacy. We deduce that group measure space II1 factors arising from actions of free groups with different number of generators are never isomorphic. We actually prove unique Cartan decomposition results for II1 factors arising from arbitrary actions of a much larger family of groups, including all free products of amenable groups and their direct products.

Free probability theory, which has became a main branch of random matrix theory, is a valuable tool for describing the asymptotic behavior of multiple systems, especially for large random matrices. In this paper, using free probability theory, a new cooperative scheme for spectrum sensing is proposed, which shows how asymptotic free behavior of random matrices and the property of Wishart distribution can be used to assist spectrum sensing for cognitive radios. Simulations over Rayleigh fading and AWGN channels demonstrate the proposed scheme has better detection performance compared with the energy detection techniques even for the case of a small sample of observations.

We study the eigenvalue distribution of random matrices pertinent to the analysis of deep neural networks. The matrices resemble the product of the sample covariance matrices, however, an important difference is that the analog of the population covariance matrix is now a function of random data matrices (synaptic weight matrices in the deep neural network terminology). The problem has been treated in recent work [J. Pennington, S. Schoenholz and S. Ganguli, The emergence of spectral universality in deep networks, Proc. Mach. Learn. Res. 84 (2018) 1924–1932, arXiv:1802.09979] by using the techniques of free probability theory. Since, however, free probability theory deals with population covariance matrices which are independent of the data matrices, its applicability in this case has to be justified. The justification has been given in [L. Pastur, On random matrices arising in deep neural networks: Gaussian case, Pure Appl. Funct. Anal. (2020), in press, arXiv:2001.06188] for Gaussian data matrices with independent entries, a standard analytical model of free probability, by using a version of the techniques of random matrix theory. In this paper, we use another version of the techniques to extend the results of [L. Pastur, On random matrices arising in deep neural networks: Gaussian case, Pure Appl. Funct. Anal. (2020), in press, arXiv:2001.06188] to the case where the entries of the data matrices are just independent identically distributed random variables with zero mean and finite fourth moment. This, in particular, justifies the mean field approximation in the infinite width limit for the deep untrained neural networks and the property of the macroscopic universality of random matrix theory in this case.

The topic of this thesis work is free probability theory. The main goal is to understand asymptotic eigenvalue distributions of large classes of random matrices. For the models discussed in [SpVa12], we obtain a quite general algorithm to plot their distributions. We also apply the tools from [BSTV14] to give a general numerical algorithm to compute the asymptotic distribution of some other types of matrix models, such as the block-linearly modified models which have been considered in [Au12,BaNe12,BaNe12b].\n\nClassical, free and non-commutative probability can be jointly understood through the combinatorics of multiplicative functions with respect to different lattices of set partitions. The second goal of this thesis is to survey on the basic topics on the combinatorics of free probability. Our basic reference is [NiSp06]. We present new results which allow to compute cumulants of products of free and Boolean independent random variables in terms of the posets of k-divisible set partitions [ArVa12]. We also find formulas relating the different types of cumulants in non-commutative probability [AHLV14].\n\nIn connection to random matrix theory, we make particular use of the combinatorial approach to operator-valued free probability ([NSS02]) to compute Cauchy-Stieljes transforms of the asymptotic eigenvalue distributions of the matrix ensembles introduced in [SpVa12]. We do this to show that our definition of a free deterministic equivalent as a concrete operator, introduced in [SpVa12], agrees with the more widespread notion of deterministic equivalents which are being used, for example, to describe recent matrix models for wireless communications [CoDe11].\n\nVoiculescu introduced free probability in 1985 in the context of operator algebras. In 1991, he found realizations of his free circular, semi-circular and Haar-unitary operators through limits of eigenvalue distributions of quite remarkable random matrix models, such as independent (self-adjoint and non-self-adjoint) Wigner and Haar-unitary random matrices. This allowed to understand Wigner's semicircle law as a special, single-variable case of a very general phenomenon on joint non-commutative distributions of large random matrices.\n\nIn 1995 he introduced operator-valued free probability, where the limiting behaviors of much more general random matrix models can be realized. A rich class of random matrix models arises from considering a polynomial P(x_1,...,x_n) in non-commutative indeterminates x_1,...,x_n,x_1^*,...,x_n^* and evaluating it on random and deterministic matrices. In this work we are specially concerned about these kind of models. We refer to them as polynomial models''.\n\nIf the inputs are (self-adjoint or non-self-adjoint) Wigner matrices, Wishart matrices, and deterministic matrices, we may consider a deterministic operator Q by evaluating P on certain operators (y_1,...,y_n) in the context of Voiculescu's free probability theory. Provided that the size of the matrices is large (but not necessarily too large), the spectral measure of the simplified model Q becomes a good approximation of the averaged eigenvalue distribution of P. The dimensions of the matrices can also be different. The free deterministic equivalent Q of P was defined in [SpVa12], based on the generalizations of [Vo91] to rectangular spaces [BG09,BG09b], and using mostly combinatorial tools from [NSS02].\n\nThe method of deterministic equivalents (DE) was introduced by Girko at the level of Cauchy-Stieltjes transforms of the considered matrix models. In contrast to DE, the simplification from P to Q can be explained very easily and does not require the polynomial to have a specific form. It will turn out that our definitions from [SpVa12] can be very effectively combined with all the elements of method described in [BMS13] for the distributions of polynomials on self-adjoint, square, asymptotically free random matrices.\n\nThroughout this work, we comment on how the different assumptions on the distributions that we input to the random matrix models affect the quality and the type of convergence of the model to its FDE. In particular, we discuss this at the combinatorial level: The different assumptions on the model determine the classes of partitions (or cumulants) that show up on the matrix sums, and hence the moments and the nature of the fixed point equations that we will get for its FDE.\n\nTo be numerically efficient one needs to understand how freeness restricts and extends to different operator-valued levels. For this, the combinatorial methods from [NSS02] are quite important.

Applications of realizations (aka linearizations) to free probability

The asymptotic behaviour of the mean spectral density for the product of N × N random unitary matrices distributed according to the Haar measure and a fixed diagonal matrix is investigated. The large N limit of the exact expression derived by Wei and Fyodorov (2008 J. Phys. A: Math. Theor. 41 502001) is calculated by a modification of the saddle point method. It is shown that in the bulk the result coincides with the one obtained within the free probability theory by Haagerup and Larsen (2000 J. Funct. Anal. 176 331). Close to the edge points of the bulk asymptotics, the mean density for large N is described by a universal function depending on a certain scaling variable. The large deviation formulae valid outside the bulk are also derived. Obtained formulae agree well with the results of direct numerical calculations.

We study of the connection between operator valued central limits for monotone, Boolean and free probability theory, which we shall call the arcsine, Bernoulli and semicircle distributions, respectively. In scalar-valued non-commutative probability these measures are known to satisfy certain arithmetic relations with respect to Boolean and free convolutions. We show that generally the corresponding operator-valued distributions satisfy the same relations only when we consider them in the fully matricial sense introduced by Voiculescu. In addition, we provide a combinatorial description in terms of moments of the operator valued arcsine distribution and we show that its reciprocal Cauchy transform satisfies a version of the Abel equation similar to the one satisfied in the scalar-valued case.

The analogues of entropy and of Fisher's information measure in free probability theory, II

We study the hydrodynamic limits of three kinds of one-dimensional stochastic log-gases known as Dyson's Brownian motion model, its chiral version, and the Bru-Wishart process studied in dynamical random matrix theory. We define the measure-valued processes so that their Cauchy transforms solve the complex Burgers-type equations. We show that applications of the method of characteristic curves to these partial differential equations provide the functional equations relating the Cauchy transforms of measures at an arbitrary time with those at the initial time. We transform the functional equations for the Cauchy transforms to those for the R-transforms and the S-transforms of the measures, which play central roles in free probability theory. The obtained functional equations for the R-transforms and the S-transforms are simpler than those for the Cauchy transforms and useful for explicit calculations including the computation of free cumulant sequences. Some of the results are argued using the notion of free convolutions.

In this paper, using free probability theory, which is a valuable tool for describing the asymptotic behavior of multiple systems, especially for large matrices, a new cooperative MIMO scheme for spectrum sensing is proposed, which shows how asymptotic free property of random matrices and the property of Wishart distribution can be used to assist spectrum sensing in MIMO system for cognitive radios. Simulations over Rayleigh fading and AWGN channels demonstrate the proposed scheme has better detection performance compared with the energy detection techniques even for the case of a small sample of observations.