Prof. Dr. Roland Speicher
Prof. Dr. Moritz Weber
Dr. Guillaume Cebron
Oberseminar zur Freien Wahrscheinlichkeit
In diesem Seminar behandeln wir Themen aus der aktuellen Forschung zur Freien Wahrscheinlichkeit.Zeit und Ort
Mittwochs, 14-16 Uhr, SR 9$L_p$-multipliers and lacunarity for compact quantum groups
In this talk I will first give a brief introduction to the Fourier multipliers on $L_p$ spaces associated to compact quantum groups, which in particular establishes some new and basic inequalities for bounded but not completely bounded multipliers on quantum groups. Then I will show that a subset of irreducible representations of the quantum group is an interpolation set of bounded $L_p$-multipliers if and only if it is a $\Lambda(p)$-set. I will also discuss the existence and construct some examples of $\Lambda(p)$-sets, particularly on free quantum groups and Drinfeld-Jimbo deformations.
Spectral distribution of the free Jacobi process
I'll briefly recall the construction of the free Jacobi process from the Hermitian matrix-valued Jacobi process. Then, I'll focus on its spectral distribution when both projections coincide. In particular, when the rank equals 1/2, this distribution is entirely determined by the that of the free unitary Brownian motion. In this respect, I'll give explicit formulas for the mixed cumulants of the latter process we obtained with M. Guay-paquet and A. Nica. For arbitrary ranks, I'll exhibit recent results toward the descrpition of the spectral distribution and present some open problems.
Products of Ginibre matrices: a finite-dimensional analysis of multi-antenna communications in presence of progressive scattering
Foreseen communication paradigms for 5G and beyond wireless cellular systems involve communication channels which, in case the transceivers are equipped with multiple antennas, can be suitably modeled by products of random matrices. While sacrificing a more accurate description of the direction of arrival of impinging and departing waves from scattering objects naturally present on the channel, which can be found in [1], the finite-size spectral analysis of the product of Ginibre matrices provides precious tools for the information- and communication-theoretic performance analysis of MIMO systems in presence of the so-called multiple cluster-scattering [2] or progressive scattering [3]. Relying on the results of [4], in this talk a two-fold analysis is carried out. First, Harish-Chandra-Itszykon-Zuber integrals are exploited to provide an expression of the Gallager Random Coding Error Exponent, which both upper bounds the Error Probability with optimal decoding at fixed coding length, and provide expressions for the Cut-Off Rate of the considered MIMO channel [5]. Then, determinantal expressions are provided for the Signal to Interference and Noise Ratio at the output of a suboptimal, multiuser receiver [6,7], in presence of a matrix-valued random interference process which can be represented by the product of Ginibre matrices, too.
Independence and Critical Scalings at Shocks in TASEP and Last Passage Percolation
We consider the totally asymmetric simple exclusion process (TASEP), an interacting particle system equivalent to a growth / last passage percolation (LPP) model which belongs to the so-called KPZ universality class. The probability laws governing the asymptotic behavior of particle positions often originate in random matrix theory and depend on the particle density. Here we consider the case when the density has a discontinuity, called shock. We show that at the shock, an asymptotic independence appears in LPP, leading to a product as limit law. Finally we consider a critical scaling, which interpolates between the product form at the shock and the constant density fluctuations.
Quantum mysteries, maximal group C*-algebras, and residual finite-dimensionality
Quantum physics is based on noncommutative probability theory. I will explain how this results in some very counterintuitive phenomena, culminating in Bell's theorem and the Kochen-Specker theorem. Measuring just how counterintuitive noncommutative probability theory can be naturally leads one to consider certain maximal group C*-algebras and C*-algebras freely generated by partitions of unity indexed by hypergraphs. Asking whether these C*-algebras are residually finite-dimensional generalizes Kirchberg's QWEP conjecture, and this question turns out to be related to the computability of the norm. This talk will be based on the following papers: http://arxiv.org/abs/1008.1168 http://arxiv.org/abs/1207.0975 http://arxiv.org/abs/1212.4084
On the structure of exotic quantum group C*-algebras
Given a compact quantum group G, there are often many ways to complete the Hopf *-algebra of polynomial functions Pol(G) to a quantum group C*-algebra. For instance, one could take the minimal C*-completion C_r(G) coming from the GNS construction for the Haar state, or the maximal completion C^u(G) by taking the universal C*-completion of Pol(G). Any quantum group C*-algebra C(G) lying as an intermediate quotient C^u(G) -> C(G) -> C_r(G) is called exotic.
In this talk, I will discuss a class of exotic quantum group C*-algebras, called L_p-C*-algebras, which are obtained by completing Pol(G) with respect to the C*-norm induced by all unitary representations of the discrete dual of G satisfying a certain L_p-integrability condition for their matrix coefficient functions (with respect to the Haar weight). In the case of free orthogonal quantum groups, it turns out that we can say a lot about these L_p C*-algebras: they are all distinct for different exponents p, they admit unique tracial states, and they fail to have any nice local properties such as local reflexivity, the local lifting property, amenable traces, or the weak expectation property.
This is based on joint work with Zhong-Jin Ruan and Matthew Wiersma.
Fourth Moment Theorems
The classical Fourth Moment Theorem says that for a normalized sequence of multiple Wiener-It\^{o} integrals, convergence of just the fourth moment suffices to ensure convergence in law towards a standard Gaussian random variable. Analogously, there also exists a Free Fourth Moment Theorem for convergence of multiple Wigner integrals towards a semicircular random variable. Both of these results were originally proven by exploiting the rich structure that multiple integrals are endowed with. In the commutative world, in an exciting new development initiated by Michel Ledoux, much more general Fourth Moment Theorems for generic eigenfunctions of Markov diffusion generators with a certain chaotic property converging towards target laws fulfilling some sufficient conditions (examples being the Gaussian, Gamma and Beta distribution) have been proved recently. We will present an overview of this new approach and some attempts to extend it to the non-commutative world.
Free Stein kernels and an improvement of the free logarithmic Sobolev inequality
In their 2015 paper, Ledoux, Nourdin, and Peccati use Stein kernels and Stein discrepancies to improve the classical logarithmic Sobolev inequality (relative to a Gaussian distribution). Simply put, Stein discrepancy measures how far a probability distribution is from the Gaussian distribution by looking at how badly it violates the integration by parts formula. In free probability, free semicircular operators are known to satisfy a corresponding 'integration by parts formula' by way of the free difference quotients. Using this fact, we define the noncommutative analogues of Stein kernels and Stein discrepancies and use them to produce an improvement of Biane and Speichers free logarithmic Sobolev inequality from 2001. We will also see several examples of free Stein kernels which have interesting connections to free monotone transport. This is based on joint work with Max Fathi.
Rational Catalan Numbers
For each rational number x=a/(b-a) where 0 < a < b are coprime integers we define a 'Catalan number' by the following formula: Cat(x):= (a+b-1)! / (a!b!).
For all positive integers n,k we note that Cat(n) are the classical Catalan numbers and Cat(n/(kn-n+1)) are the classical Fuss-Catalan numbers.
In the classical cases the Catalan numbers are well-known to count various kinds of noncrossing partitions and polygon triangulations. I will present a generalization of these constructions to the rational case. If we define the 'rational q-Catalan number' by
Cat_q(x):= [a+b-1]! / ([a]![b]!),
where [n]! is the q-factorial, then the usual 'cyclic sieving' properties of partitions and triangulations also generalize to the rational case.
Finally, I will mention some non-classical phenomena that emerge only at the rational level of generality.
On noncommutative distributional symmetries and de Finetti theorems associated with them
I will present a recent work on general de Finetti theorems for classical, free and boolean independencies. I will give an application of our theorems to all easy quantum groups and easy groups
De Finetti theorems and bi-freeness
I will report on a joint work with M. Weber on possible generalizations of the free de Finetti theorem to the setting of bi-free probability. I will first recall the combinatorics of bi-freenes and then explain the problems which arise when one tries to prove de Finetti theorems in that context.
Realizations of noncommutative rational functions and minimization
After introducing briefly the notion of noncommutative rational expressions and functions, we will focus on the theory of 'realizations'. A realization represents a noncommutative rational function as a linear combination of entries of an inverse of a matrix, whose entries are linear polynomials. In particular, we will see that there exist algorithms to produce such realizations. Since a realization of a noncommutative rational function is not unique, it is natural to search for realizations with matrices of minimal size. In the case where the noncommutative rational function is regular at zero, there exists an explicit algorithm for cutting down any given realization to a minimal one. Finally, we will discuss recent results of J. Volcic, generalizing this to the non-regular case.
Cumulants for Finite Free Convolution
In recent paper, Marcus introduced a 'finite version' of Free Convolution which is based on convolution of polynomials. The limit as the degree of the polynomials tend to infinity coincides with free convolution. In this talk I will describe cumulants for this finite free convolution and show that as expected they approximate free cumulants.
Universality of the limiting spectral distribution for matrices with correlated entries
We are in the asymptotic spectral behaviour of random matrices having correlated entries that are functions of i.i.d. random variables. We show that the limiting spectral distribution can be obtained by analysing a Gaussian matrix having the same covariance structure. This approximation approach is valid for both short and long range dependent stationary random processes just having moments of second order. Our approach is based on a blend of a blocking procedure, Lindeberg’s method and the Gaussian interpolation technique. (joint work with F. Merlevède and M. Peligrad)
| Aktualisiert am: 7. September 2016 Miguel Pluma | Impressum |
