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      Free Probability and Random Matrices



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ERC Advanced Grant

From 1 February 2014 until 31 January 2019, the European Research Council supported the research of Roland Speicher by an ERC Advanced Grant with the title Non-Commutative Distributions in Free Probability. More info on this can be found here


Symbolic Tools in Mathematics and their Application

The Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) has established the transregional collaborative research centre (SFB-TRR) 195 “Symbolic Tools in Mathematics and their Application” starting in January 2017.  Host university is the University of Kaiserslautern (TU Kaiserslautern), additional applicant universities are RWTH Aachen University and Saarland University.

What actually is Free Probability?

A motivating example

Consider the following three natural questions from the fields of operator algebras, random matrices, and combinatorics, respectively.

  • What is the distribution of the real part of the one-sided shift with respect to the canonical vacuum expectation state?
  • What is the asymptotic eigenvalue distribution of a Gaussian random matrix in the limit of large matrix size?
  • In how many ways can one connect 2n points on a circle pairwisely by n non-intersecting chords?

Surprisingly, all three questions have essentially the same answer: both question 1 and 2 yield the semicircular distribution, and the answer to question 3 is given by the Catalan numbers which are actually the even moments of the semicircular distribution. This coincidence is not just superficial - a conceptual reason is given by Voiculescu's free probability theory and its basic concept of freeness. [...]

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