arXiv subjects

Roland Speicher

Publications and source records attributed to Roland Speicher.

At least 73 records · Page 4Linked to original sources

arXiv2005

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 derive 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.

math.OA↗

arXiv2005

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.

math.OA↗

arXiv2005

Orthogonal Polynomials and Fluctuations of Random Matrices

In this paper we establish a connection between the fluctuations of Wishart random matrices, shifted Chebyshev polynomials, and planar diagrams whose linear span form a basis for the irreducible representations of the annular Temperly-Lieb algebra.

math.OA↗

arXiv2004

On the structure of non-commutative white noises

We consider the concepts of continuous Bernoulli systems and non-commutative white noises. We address the question of isomorphism of continuous Bernoulli systems and show that for large classes of quantum L{é}vy processes one can make quite precise statements about the time behaviour of their moments.

math.OA↗

arXiv2001

Operator-valued distributions: I. Characterizations of freeness

Let $M$ be a $B$-probability space. Assume that $B$ itself is a $D$-probability space; then $M$ can be viewed as $D$-probability space as well. Let $X$ be in $M$. We look at the question of relating the properties of $X$ as $B$-valued random variable to its properties as $D$-valued random variable. We characterize freeness of $X$ from $B$ with amalgamation over $D$: (a) in terms of a certain factorization condition linking the $B$-valued and $D$-valued cumulants of $X$, and (b) for $D$ finite-dimensional, in terms of linking the $B$-valued and the $D$-valued Fisher information of $X$. We give an application to random matrices. For the second characterization we derive a new operator-valued description of the conjugate variable and introduce an operator-valued version of the liberation gradient.

math.OA↗

arXiv2001

Free Calculus

I give a survey about my work on combinatorial and probabilistic aspects of free probability theory. In particular, I present the combinatorial description of freeness in terms of free cumulants and I give some ideas of the main results of my joint work with Philippe Biane on a stochastic integration theory for free Brownian motion.

math.OA↗

arXiv2001

Continuous Family of Invariant Subspaces for R-diagonal Operators

We show that every R-diagonal operator x has a continuous family of invariant subspaces relative to the von Neumann algebra generated by x. This allows us to find the Brown measure of x and to find a new conceptual proof that Voiculescu's S-transform is multiplicative. Our considerations base on a new concept of R-diagonality with amalgamation, for which we give several equivalent characterizations.

math.FA↗

arXiv2001

A Characterization of Freeness by a Factorization Property of R-transform

Let M be a B-probability space. Assume that B itself is a D-probability space; then M can be viewed as a D-probability space as well. Let X be in M. We characterize freeness of X from B with amalgamation over D in terms of a certain factorization condition linking the B-valued and D-valued R-transforms of X. We give an application to random matrices.

math.OA↗

arXiv2001

R-cyclic families of matrices in free probability

We introduce the concept of ``R-cyclic family'' of matrices with entries in a non-commutative probability space; the definition consists in asking that only the ``cyclic'' non-crossing cumulants of the entries of the matrices are allowed to be non-zero. Let A_{1}, ..., A_{s} be an R-cyclic family of d \times d matrices over a non-commutative probability space. We prove a convolution-type formula for the explicit computation of the joint distribution of A_{1}, ..., A_{s} (considered in M_{d} (\A) with the natural state), in terms of the joint distribution (considered in the original space) of the entries of the s matrices. Several important situations of families of matrices with tractable joint distributions arise by application of this formula. Moreover, let A_{1}, ..., A_{s} be a family of d \times d matrices over a non-commutative probability space, let \D \subset M_{d} (\A) denote the algebra of scalar diagonal matrices, and let {\cal C} be the subalgebra of M_{d} (\A) generated by \{A_{1}, ..., A_{s} \} \cup \D. We prove that the R-cyclicity of A_{1}, ..., A_{s} is equivalent to a property of {\cal C} -- namely that {\cal C} is free from M_{d} (\C), with amalgamation over \D.

math.OA↗

arXiv1999

Combinatorics of free cumulants

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.

math.CO↗

arXiv1996

Commutators of free random variables

Let A be a unital $C^*$-algebra, given together with a specified state $ϕ:A \to C$. Consider two selfadjoint elements a,b of A, which are free with respect to $ϕ$ (in the sense of the free probability theory of Voiculescu). Let us denote $c:=i(ab-ba)$, where the i in front of the commutator is introduced to make c selfadjoint. In this paper we show how the spectral distribution of c can be calculated from the spectral distributions of a and b. Some properties of the corresponding operation on probability measures are also discussed. The methods we use are combinatorial, based on the description of freeness in terms of non-crossing partitions; an important ingredient is the notion of R-diagonal pair, introduced and studied in our previous paper funct-an/9604012.

funct-an↗

arXiv1996

On the multiplication of free $n$-tuples of non-commutative random variables

Let $a_{1},...,a_{n}, b_{1},...,b_{n}$ be random variables in some (non-commutative) probability space, such that $\{a_{1}, ..., a_{n} \}$ is free from $\{b_{1}, ..., b_{n} \}$. We show how the joint distribution of the $n$-tuple $(a_{1} b_{1}, ..., a_{n} b_{n})$ can be described in terms of the joint distributions of $(a_{1}, ..., a_{n})$ and $(b_{1}, >..., b_{n})$, by using the combinatorics of the $n$-dimensional $R$-transform. We point out a few applications that can be easily derived from our result, concerning the left-and-right translation with a semicircular element (see Sections 1.6-1.10) and the compression with a projection (see Sections 1.11-1.14) of an $n$-tuple of non-commutative random variables. A different approach to two of these applications is presented by Dan Voiculescu in an Appendix to the paper.

funct-an↗

arXiv1996

$R$-diagonal pairs - a common approach to Haar unitaries and circular elements

In the free probability theory of Voiculescu two of the most frequently used *-distributions are those of a Haar unitary and of a circular element. We define an $R$-diagonal pair as a generalization of these distributions by the requirement that their two-dimensional $R$-transform (or free cumulants) have a special diagonal form. We show that the class of such $R$-diagonal pairs has an absorption property under nested multiplication of free pairs. This implies that in the polar decomposition of such an element the polar part and the absolute value are free. Our calculations are based on combinatorial statements about non-crossing partitions, in particular on a canonical bijection between the set of intervals of NC(n) and the set of 2-divisible partitions in NC(2n). In a forthcoming paper the theory of $R$-diagonal pairs will be used to solve the problem of the free commutator.

funct-an↗

arXiv1996

$q$-Gaussian processes: non-commutative and classical aspects

We examine, for $-1<q<1$, $q$-Gaussian processes, i.e. families of operators (non-commutative random variables) $X_t=a_t+a_t^*$ -- where the $a_t$ fulfill the $q$-commutation relations $a_sa_t^*-qa_t^*a_s=c(s,t)\cdot \id$ for some covariance function $c(\cdot,\cdot)$ -- equipped with the vacuum expectation state. We show that there is a $q$-analogue of the Gaussian functor of second quantization behind these processes and that this structure can be used to translate questions on $q$-Gaussian processes into corresponding (and much simpler) questions in the underlying Hilbert space. In particular, we use this idea to show that a large class of $q$-Gaussian processes possess a non-commutative kind of Markov property, which ensures that there exist classical versions of these non-commutative processes. This answers an old question of Frisch and Bourret \cite{FB}.

funct-an↗

arXiv1994

Rigorous mean field model for CPA: Anderson model with free random variables

A model of a randomly disordered system with site-diagonal random energy fluctuations is introduced. It is an extension of Wegner's $n$-orbital model to arbitrary eigenvalue distribution in the electronic level space. The new feature is that the random energy values are not assumed to be independent at different sites but free. Freeness of random variables is an analogue of the concept of independence for non-commuting random operators. A possible realization is the ensemble of at different lattice-sites randomly rotated matrices. The one- and two-particle Green functions of the proposed hamiltonian are calculated exactly. The eigenstates are extended and the conductivity is nonvanishing everywhere inside the band. The long-range behaviour and the zero-frequency limit of the two-particle Green function are universal with respect to the eigenvalue distribution in the electronic level space. The solutions solve the CPA-equation for the one- and two-particle Green function of the corresponding Anderson model. Thus our (multi-site) model is a rigorous mean field model for the (single-site) CPA. We show how the Llyod model is included in our model and treat various kinds of noises.

cond-mat↗

arXiv1994

Random matrix theory for CPA: Generalization of Wegner's $n$--orbital model

We introduce a generalization of Wegner's $n$-orbital model for the description of randomly disordered systems by replacing his ensemble of Gaussian random matrices by an ensemble of randomly rotated matrices. We calculate the one- and two-particle Green's functions and the conductivity exactly in the limit $n\to\infty$. Our solution solves the CPA-equation of the $(n=1)$-Anderson model for arbitrarily distributed disorder. We show how the Lloyd model is included in our model.

cond-mat↗

arXiv1994

Convolution and Limit Theorems for Conditionally Free Random Variables

We introduce the notion of a conditionally free product and conditionally free convolution. We describe this convolution both from a combinatorial point of view, by showing its connection with the lattice of non-crossing partitions, and from an analytic point of view, by presenting the basic formula for its $R$-transform. We calculate explicitly the distributions of the conditionally free Gaussian and conditionally free Poisson distribution.

funct-an↗

arXiv1994

Interpolations between Bosonic and Fermionic Relations given by Generalized Brownian Motions (revised version)

We present an interpolation between the bosonic and fermionic relations. This interpolation is given by an object which we call `generalized Brownian motion' and which is characterized by a generalization of the pairing rule for the calculation of the moments of bosonic and fermionic fields. We develop some basic theory for such generalized Brownian motions and consider more closely one example, which turns out to be intimately connected with Voiculescu's concept of `free product'.

funct-an↗