# Operator-limit distributions in probability theory

## Probability Surveys - Vol. 14 ()

The main directions are: i classifying non-commutative symmetries and describing the effect of invariance under such quantum symmetries for non-commutative distributions; this will rely on our recent theory of easy quantum groups ii proving regularity properties for non-commutative distributions; for this we will develop the theory of free Malliavin calculus iii providing algorithms for calculating non-commutative distributions; this will rely on advances of the analytic theory of operator valued free convolutions and will in particular lead to a master algorithm for the computation of asymptotic eigenvalue distributions for general random matrix problems".

Activity type Higher or Secondary Education Establishments. Website Contact the organisation. Principal Investigator Roland Speicher Prof.

Administrative Contact Corinna Hahn Ms. Status Closed project. Start date 1 February End date 31 January Typical examples of such operators are non-commuting random matrices. A main quantity of interest is to understand the eigenvalue distribution of various functions built out of those random matrices; in particular, when the size of the matrices gets large. Such questions, though a priori of theoretical nature, have quite some relevance for concrete applications, like in wireless networks or machine learning, where random matrices are used to model the transmission channels, and the eigenvalue distribution contains important information about the performance of the considered systems.

The non-commutativity of the involved operators moves this problem out of the realm of classical methods. Classical probability theory looks on similar questions, but since there the involved random variables commute, their distribution is a classical probability measure - a well-understood object, for which there exist many far-developed tools to study it.

The non-commutative counterpart, which we study here, is much harder to grasp and we are in need of totally new methods to get an understanding of non-commutative distributions.

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In the ERC grant we made several break-throughs in such a theory of non-commutative distributions. One direction was to understand non-commutative distributions by their symmetries. Such quantum groups have been studied for quite a while by now in mathematics, but we were able to find, describe and classify new classes of such quantum groups, which are of great relevance for non-commutative distributions.

## Conditional Probability Theory (For Scientists and Engineers)

Furthermore, we developed new analytic tools for deriving qualitative features of non-commutative distributions in a very general setting and we constructed and implemented new algorithms for the calculation of non-commutative distributions. Those computation algorithms go much beyond what was available before and will be instrumental in virtually any field where the asymptotic eigenvalue distribution of random matrices plays a role.

Deliverables Deliverables not available. Publications Publications via OpenAire. Mallows Permutations and Finite Dependence. Geometric Ergodicity in a Weighted Sobolev Space. Hydrodynamics in a condensation regime: the disordered asymmetric zero-range process. The maximal flow from a compact convex subset to infinity in first passage percolation on Z d.

Hitting probabilities of a Brownian flow with Radial Drift. Random Moment Problems Under Constraints. On the probability of nonexistence in binomial subsets. The maximum of the four-dimensional membrane model. Cutoff for the mean-field zero-range process with bounded monotone rates. Translation-invariant Gibbs states of the Ising model: general setting. The endpoint distribution of directed polymers. The distribution of Gaussian multiplicative chaos on the unit interval.

The two-dimensional KPZ equation in the entire subcritical regime. Exchangeable interval hypergraphs and limits of ordered discrete structures. On the topological boundary of the range of super-Brownian motion. Entrance and exit at infinity for stable jump diffusions.

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Ergodic Poisson Splittings. Operator limit of the circular beta ensemble. Identification of the Polaron measure in strong coupling and the Pekar variational formula. Entropic repulsion for the occupation-time field of random interlacements conditioned on disconnection. Flows, coalescence and noise. A correction. Locality of the critical probability for transitive graphs of exponential growth. Percolation for level-sets of Gaussian free fields on metric graphs. Large deviations for the largest eigenvalue of Rademacher matrices. Planar Brownian motion and Gaussian multiplicative chaos.

### The Annals of Mathematical Statistics

Bounding the number of self-avoiding walks: Hammersley-Welsh with polygon insertion. Probability theory and related fields 78 3 , , Journal of Multivariate Analysis 13 4 , , Annales Univer. Articles 1—20 Show more. Help Privacy Terms. An integral representation of operator-selfdecomposable random variables ZJ Jurek Bull.

Limit distributions and one-parameter groups of linear operators on Banach spaces ZJ Jurek Journal of Multivariate Analysis 13 4 , , Polish Acad. Math 31, , Math 28, , Math 32, , Series of independent exponential random variables ZJ Jurek Proc.