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Roland Speicher

Publications and source records attributed to Roland Speicher.

At least 19 recordsLinked to original sources
arXiv2025

Lecture Notes on "Free Probability Theory"

This in an introduction to free probability theory, covering the basic combinatorial and analytic theory, as well as the relations to random matrices and operator algebras. The material is mainly based on the two books of the lecturer, one joint with Nica and one joint with Mingo. Free probability is here restricted to the scalar-valued setting, the operator-valued version is treated in the subsequent lecture series on "Non-Commutative Distributions". The material here was presented in the winter term 2018/19 at Saarland University in 26 lectures of 90 minutes each. The lectures were recorded and can be found online at https://www.youtube.com/playlist?list=PLY11JnnnTUCYZni2Q7QNVa9hPGu77GK4M

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arXiv2024

Entrywise application of non-linear functions on orthogonally invariant matrices

In this article, we investigate how the entrywise application of a non-linear function to symmetric orthogonally invariant random matrix ensembles alters the spectral distribution. We treat also the multivariate case where we apply multivariate functions to entries of several orthogonally invariant matrices; where even correlations between the matrices are allowed. We find that in all those cases a Gaussian equivalence principle holds, that is, the asymptotic effect of the non-linear function is the same as taking a linear combination of the involved matrices and an additional independent GOE. The ReLU-function in the case of one matrix and the max-function in the case of two matrices provide illustrative examples.

math.SP↗
arXiv2024

Computing the noncommutative inner rank by means of operator-valued free probability theory

We address the noncommutative version of the Edmonds' problem, which asks to determine the inner rank of a matrix in noncommuting variables. We provide an algorithm for the calculation of this inner rank by relating the problem with the distribution of a basic object in free probability theory, namely operator-valued semicircular elements. We have to solve a matrix-valued quadratic equation, for which we provide precise analytical and numerical control on the fixed point algorithm for solving the equation. Numerical examples show the efficiency of the algorithm.

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arXiv2022

A dual and conjugate system for $q$-Gaussians for all $q$

We provide a concrete formula for a dual system as well as for a conjugate system of $q$-Gaussians represented on the $q$-deformed Fock space. Moreover, using this formula, we prove the existence of a free Gibbs potential and that the non-microstates free Fisher information is finite for any $q$ with $-1<q<1$, which is an improvement on a previous result of Y. Dabrowski. We also indicate how our results can be extended to the more general setting of mixed $q_{ij}$-relations.

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arXiv2022

Universality of free random variables: atoms for non-commutative rational functions

Consider a tuple $(Y_1,\dots,Y_d)$ of normal operators in a tracial operator algebra setting with prescribed sizes of the eigenspaces for each $Y_i$. We address the question what one can say about the sizes of the eigenspaces for any non-commutative polynomial $P(Y_1,\dots,Y_d)$ in those operators? We show that for each polynomial $P$ there are unavoidable eigenspaces, which occur in $P(Y_1,\dots,Y_d)$ for any $(Y_1,\dots,Y_d)$ with the prescribed eigenspaces for the marginals. We will describe this minimal situation both in algebraic terms - where it is given by realizations via matrices over the free skew field and via rank calculations - and in analytic terms - where it is given by freely independent random variables with prescribed atoms in their distributions. The fact that the latter situation corresponds to this minimal situation allows to draw many new conclusions about atoms in polynomials of free variables. In particular, we give a complete description of atoms in the free commutator and the free anti-commutator. Furthermore, our results do not only apply to polynomials, but much more general also to non-commutative rational functions. Since many random matrix models become asymptotically free in the large $N$ limit, our results allow us to calculate the location and size of atoms in the asymptotic eigenvalue distribution of polynomials and rational functions in randomly rotated matrices.

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arXiv2020

Joint Global Fluctuations of complex Wigner and deterministic Matrices

We characterize the limiting fluctuations of traces of several independent Wigner matrices and deterministic matrices under mild conditions. A CLT holds but in general the families are not asymptotically free of second order and the limiting covariance depends on more information on the deterministic matrices than their limiting *-distribution.

math.PR↗
arXiv2020

Lecture Notes on "Random Matrices"

This in an introduction to random matrix theory, giving an impression of some of the most important aspects of this modern subject. In particular, it covers the basic combinatorial and analytic theory around Wigner's semicircle law, featuring also concentration phenomena, and the Tracy-Widom distribution of the largest eigenvalue. The circular law and a discussion of Voiculescu's multivariate extension of the semicircle law, as an appetizer for free probability theory, also make an appearance. The material here was presented in the winter term 2019/20 at Saarland University in 24 lectures of 90 minutes each. The lectures were recorded and can be found online at https://www.math.uni-sb.de/ag/speicher/web_video/index.html.

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arXiv2020

A dynamical version of the SYK Model and the q-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.

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arXiv2020

Lecture Notes on "Non-Commutative Distributions"

This in an introduction to the theory of non-commutative distributions of non-commuting operators or random matrices. Starting from the basic problem to find a good approach to the meaning of "non-commutative distribution" we will, in particular, cover: free analysis, which is a version of complex analysis for several non-commuting variables; the operator-valued version of free probability theory (combinatorial but also analytic aspects); the linearization trick to reduce non-linear scalar problems to linear operator-valued problems; the combination of operator-valued convolution and linearization to calculate the distribution of polynomials in free variables; the basic theory of non-commutative rational functions. On one hand, this is a continuation of the Lecture Notes on "Free Probability", arXiv:1908.08125. On the other hand, the theory of free probability is developed again, but in a more general, operator-valued context. So, in principle and with some additional efforts, it should be possible to read the present notes without having a prior knowledge on free probability. Big parts of the material do also not deal so much with free variables, but more general with analytic and algebraic aspects of maximal non-commuting variables. The material here was presented in the summer term 2019 at Saarland University in 20 lectures of 90 minutes each. The lectures were recorded and can be found online at https://www.math.uni-sb.de/ag/speicher/web_video/index.html.

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arXiv2020

A Note on the Free and Cyclic Differential Calculus

In 2000, Voiculescu proved an algebraic characterization of cyclic gradients of noncommutative polynomials. We extend this remarkable result in two different directions: first, we obtain an analogous characterization of free gradients; second, we lift both of these results to Voiculescu's fundamental framework of multivariable generalized difference quotient rings. For that purpose, we develop the concept of divergence operators, for both free and cyclic gradients, and study the associated (weak) grading and cyclic symmetrization operators, respectively. One the one hand, this puts a new complexion on the initial polynomial case, and on the other hand, it provides a uniform framework within which also other examples - such as a discrete version of the Itô stochastic integral - can be treated.

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arXiv2020

The free field: zero divisors, Atiyah property and realizations via unbounded operators

We consider noncommutative rational functions as well as matrices in polynomials in noncommuting variables in two settings: in an algebraic context the variables are formal variables, and their rational functions generate the "free field"; in an analytic context the variables are given by operators from a finite von Neumann algebra and the question of rational functions is treated within the affiliated unbounded operators. Our main result shows that for a "good" class of operators - namely those for which the free entropy dimension is maximal - the analytic and the algebraic theory are isomorphic. This means in particular that any non-trivial rational function can be evaluated as an unbounded operator for any such good tuple and that those operators don't have zero divisors. On the matrix side, this means that matrices of polynomials which are invertible in the free field are also invertible as matrices over unbounded operators when we plug in our good operator tuples. We also address the question how this is related to the strong Atiyah property. The above yields a quite complete picture for the question of zero divisors (or atoms in the corresponding distributions) for operator tuples with maximal free entropy dimension. We give also some partial results for the question of existence and regularity of a density of the distribution.

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arXiv2020

The free field: realization via unbounded operators and Atiyah property

Let $X_1,\dots,X_n$ be operators in a finite von Neumann algebra and consider their division closure in the affiliated unbounded operators. We address the question when this division closure is a skew field (aka division ring) and when it is the free skew field. We show that the first property is equivalent to the strong Atiyah property and that the second property can be characterized in terms of the non-commutative distribution of $X_1,\dots,X_n$. More precisely, $X_1,\dots,X_n$ generate the free skew field if and only if there exist no non-zero finite rank operators $T_1,\dots,T_n$ such that $\sum_i[T_i,X_i]=0$. Sufficient conditions for this are the maximality of the free entropy dimension or the existence of a dual system of $X_1,\dots,X_n$. Our general theory is not restricted to selfadjoint operators and thus does also include and recover the result of Linnell that the generators of the free group give the free skew field. We give also consequences of our result for the question of atoms in the distribution of rational functions in free variables or in the asymptotic eigenvalue distribution of matrices over polynomials in asymptotically free random matrices. This solves in particular a conjecture of Charlesworth and Shlyakhtenko.

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arXiv2017

Applications of Realizations (aka Linearizations) to Free Probability

We show how the combination of new "linearization" ideas in free probability theory with the powerful "realization" machinery -- developed over the last 50 years in fields including systems engineering and automata theory -- allows solving the problem of determining the eigenvalue distribution (or even the Brown measure, in the non-selfadjoint case) of noncommutative rational functions of random matrices when their size tends to infinity. Along the way we extend evaluations of noncommutative rational expressions from matrices to stably finite algebras, e.g. type II$_1$ von Neumann algebras, with a precise control of the domains of the rational expressions. The paper provides sufficient background information, with the intention that it should be accessible both to functional analysts and to algebraists.

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arXiv2017

Eigenvalues of non-hermitian random matrices and Brown measure of non-normal operators: hermitian reduction and linearization method

We study the Brown measure of certain non-hermitian operators arising from Voiculescu's free probability theory. Usually those operators appear as the limit in *-moments of certain ensembles of non-hermitian random matrices, and the Brown measure gives then a canonical candidate for the limit eigenvalue distribution of the random matrices. A prominent class for our operators is given by polynomials in *-free variables. Other explicit examples include R-diagonal elements and elliptic elements, for which the Brown measure was already known, and a new class of triangular-elliptic elements. Our method for the calculation of the Brown measure is based on a rigorous mathematical treatment of the hermitian reduction method, as considered in the physical literature, combined with subordination ideas and the linearization trick.

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