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Roland Speicher

Publications and source records attributed to Roland Speicher.

At least 55 records · Page 3Linked to original sources
arXiv2010

The normal distribution is $\boxplus$-infinitely divisible

We prove that the classical normal distribution is infinitely divisible with respect to the free additive convolution. We study the Voiculescu transform first by giving a survey of its combinatorial implications and then analytically, including a proof of free infinite divisibility. In fact we prove that a subfamily Askey-Wimp-Kerov distributions are freely infinitely divisible, of which the normal distribution is a special case. At the time of this writing this is only the third example known to us of a nontrivial distribution that is infinitely divisible with respect to both classical and free convolution, the others being the Cauchy distribution and the free 1/2-stable distribution.

math.OA↗
arXiv2009

Asymptotic infinitesimal freeness with amalgamation for Haar quantum unitary random matrices

We consider the limiting distribution of $U_NA_NU_N^*$ and $B_N$ (and more general expressions), where $A_N$ and $B_N$ are $N \times N$ matrices with entries in a unital C$^*$-algebra $\mathcal B$ which have limiting $\mathcal B$-valued distributions as $N \to \infty$, and $U_N$ is a $N \times N$ Haar distributed quantum unitary random matrix with entries independent from $\mathcal B$. Under a boundedness assumption, we show that $U_NA_NU_N^*$ and $B_N$ are asymptotically free with amalgamation over $\mathcal B$. Moreover, this also holds in the stronger infinitesimal sense of Belinschi-Shlyakhtenko. We provide an example which demonstrates that this example may fail for classical Haar unitary random matrices when the algebra $\mathcal B$ is infinite-dimensional.

math.OA↗
arXiv2009

The Non-Commutative Cycle Lemma

We present a non-commutative version of the cycle lemma of Dvoretsky and Motzkin that applies to free groups and use this result to solve a number of problems involving cyclic reduction in the free group. We also describe an application to random matrices, in particular the fluctuations of Kesten's Law.

math.CO↗
arXiv2009

Free Probability Theory

Free probability theory was created by Dan Voiculescu around 1985, motivated by his efforts to understand special classes of von Neumann algebras. His discovery in 1991 that also random matrices satisfy asymptotically the freeness relation transformed the theory dramatically. Not only did this yield spectacular results about the structure of operator algebras, but it also brought new concepts and tools into the realm of random matrix theory. In the following we will give, mostly from the random matrix point of view, a survey on some of the basic ideas and results of free probability theory.

math.PR↗
arXiv2009

Sharp Bounds for Sums Associated to Graphs of Matrices

We provide a simple algorithm for finding the optimal upper bound for sums of products of matrix entries of the form S_pi(N) := sum_{j_1, ..., j_2m = 1}^N t^1_{j_1 j_2} t^2_{j_3 j_4} ... t^m_{j_2m-1 j_2m} where some of the summation indices are constrained to be equal. The upper bound is easily obtained from a graph G associated to the constraints in the sum.

math.OA↗
arXiv2009

Stochastic aspects of easy quantum groups

We consider several orthogonal quantum groups satisfying the easiness assumption axiomatized in our previous paper. For each of them we discuss the computation of the asymptotic law of Tr(u^k) with respect to the Haar measure, u being the fundamental representation. For the classical groups O_n, S_n we recover in this way some well-known results of Diaconis and Shahshahani.

math.OA↗
arXiv2009

Liberation of orthogonal Lie groups

We show that under suitable assumptions, we have a one-to-one correspondence between classical groups and free quantum groups, in the compact orthogonal case. We classify the groups under correspondence, with the result that there are exactly 6 of them: $O_n,S_n,H_n,B_n,S_n',B_n'$. We investigate the representation theory aspects of the correspondence, with the result that for $O_n,S_n,H_n,B_n$, this is compatible with the Bercovici-Pata bijection. Finally, we discuss some more general classification problems in the compact orthogonal case, notably with the construction of a new quantum group.

math.QA↗
arXiv2009

Statistical eigen-inference from large Wishart matrices

We consider settings where the observations are drawn from a zero-mean multivariate (real or complex) normal distribution with the population covariance matrix having eigenvalues of arbitrary multiplicity. We assume that the eigenvectors of the population covariance matrix are unknown and focus on inferential procedures that are based on the sample eigenvalues alone (i.e., "eigen-inference"). Results found in the literature establish the asymptotic normality of the fluctuation in the trace of powers of the sample covariance matrix. We develop concrete algorithms for analytically computing the limiting quantities and the covariance of the fluctuations. We exploit the asymptotic normality of the trace of powers of the sample covariance matrix to develop eigenvalue-based procedures for testing and estimation. Specifically, we formulate a simple test of hypotheses for the population eigenvalues and a technique for estimating the population eigenvalues in settings where the cumulative distribution function of the (nonrandom) population eigenvalues has a staircase structure. Monte Carlo simulations are used to demonstrate the superiority of the proposed methodologies over classical techniques and the robustness of the proposed techniques in high-dimensional, (relatively) small sample size settings. The improved performance results from the fact that the proposed inference procedures are "global" (in a sense that we describe) and exploit "global" information thereby overcoming the inherent biases that cripple classical inference procedures which are "local" and rely on "local" information.

math.ST↗
arXiv2008

Resolvents of R-Diagonal Operators

We consider the resolvent $(λ-a)^{-1}$ of any $R$-diagonal operator $a$ in a $\mathrm{II}_1$-factor. Our main theorem gives a universal asymptotic formula for the norm of such a resolvent. En route to its proof, we calculate the $R$-transform of the operator $|λ-c|^2$ where $c$ is Voiculescu's circular operator, and give an asymptotic formula for the negative moments of $|λ-a|^2$ for any $R$-diagonal $a$. We use a mixture of complex analytic and combinatorial techniques, each giving finer information where the other can give only coarse detail. In particular, we introduce {\em partition structure diagrams}, a new combinatorial structure arising in free probability.

math.OA↗
arXiv2008

A noncommutative de Finetti theorem: Invariance under quantum permutations is equivalent to freeness with amalgamation

We show that the classical de Finetti theorem has a canonical noncommutative counterpart if we strengthen `exchangeability' (i.e., invariance of the joint distribution of the random variables under the action of the permutation group) to invariance under the action of the quantum permutation group. More precisely, for an infinite sequence of noncommutative random variables, we prove that invariance of their joint distribution under quantum permutations is equivalent to the fact that the random variables are identically distributed and free with respect to the conditional expectation onto their tail algebra.

math.OA↗
arXiv2007

Second Order Cumulants of products

We derive a formula which expresses a second order cumulant whose entries are products as a sum of cumulants where the entries are single factors. This extends to the second order case the formula of Krawczyk and Speicher. We apply our result to the problem of calculating the second order cumulants of a semi-circular and Haar unitary operator.

math.OA↗
arXiv2007

On slow-fading non-separable correlation MIMO systems

In a frequency selective slow-fading channel in a MIMO system, the channel matrix is of the form of a block matrix. We propose a method to calculate the limit of the eigenvalue distribution of block matrices if the size of the blocks tends to infinity. We will also calculate the asymptotic eigenvalue distribution of $HH^*$, where the entries of $H$ are jointly Gaussian, with a correlation of the form $E[h_{pj}\bar h_{qk}]= \sum_{s=1}^t Ψ^{(s)}_{jk}\hatΨ^{(s)}_{pq}$ (where $t$ is fixed and does not increase with the size of the matrix). We will use an operator-valued free probability approach to achieve this goal. Using this method, we derive a system of equations, which can be solved numerically to compute the desired eigenvalue distribution.

cs.IT↗
arXiv2007

Operator-valued semicircular elements: Solving a quadratic matrix equation with positivity constraints

We show that the quadratic matrix equation $VW + η(W)W = I$, for given $V$ with positive real part and given analytic mapping $η$ with some positivity preserving properties, has exactly one solution $W$ with positive real part. We point out the relevance of this result in the context of operator-valued free probability theory and for the determination of the asymptotic eigenvalue distribution of band or block random matrices. We also address the problem of a numerical determination of the solution.

math.OA↗
arXiv2006

Spectra of large block matrices

In a frequency selective slow-fading channel in a MIMO system, the channel matrix is of the form of a block matrix. This paper proposes a method to calculate the limit of the eigenvalue distribution of block matrices if the size of the blocks tends to infinity. While it considers random matrices, it takes an operator-valued free probability approach to achieve this goal. Using this method, one derives a system of equations, which can be solved numerically to compute the desired eigenvalue distribution. The paper initially tackles the problem for square block matrices, then extends the solution to rectangular block matrices. Finally, it deals with Wishart type block matrices. For two special cases, the results of our approach are compared with results from simulations. The first scenario investigates the limit eigenvalue distribution of block Toeplitz matrices. The second scenario deals with the distribution of Wishart type block matrices for a frequency selective slow-fading channel in a MIMO system for two different cases of $n_R=n_T$ and $n_R=2n_T$. Using this method, one may calculate the capacity and the Signal-to-Interference-and-Noise Ratio in large MIMO systems.

cs.IT↗
arXiv2006

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 R-transform formulas for second order freeness. Much of the presented theory relies on a detailed study of the properties of "partitioned permutations".

math.OA↗
arXiv2005

Strong Haagerup inequalities for free R-diagonal elements

In this paper, we generalize Haagerup's inequality (on convolution norm in the free group) to a very general context of R-diagonal elements in a tracial von Neumann algebra; moreover, we show that in this "holomorphic" setting, the inequality is greatly improved from its originial form. We give an elementary combinatorial proof of a very special case of our main result, and then generalize these techniques. En route, we prove a number of moment and cumulant estimates for R-diagonal elements that are of independent interest. Finally, we use our strong Haagerup inequality to prove a strong ultracontractivity theorem.

math.OA↗