Free probability

Short description: Mathematical theory on random variables

Free probability is a mathematical theory that studies non-commutative random variables. The "freeness" or free independence property is the analogue of the classical notion of independence, and it is connected with free products.

This theory was initiated by Dan Voiculescu around 1986 in order to attack the free group factors isomorphism problem, an important unsolved problem in the theory of operator algebras. Given a free group on some number of generators, we can consider the von Neumann algebra generated by the group algebra, which is a type II1 factor. The isomorphism problem asks whether these are isomorphic for different numbers of generators. It is not even known if any two free group factors are isomorphic. This is similar to Tarski's free group problem, which asks whether two different non-abelian finitely generated free groups have the same elementary theory.

Later connections to random matrix theory, combinatorics, representations of symmetric groups, large deviations, quantum information theory and other theories were established. Free probability is currently undergoing active research.

Typically the random variables lie in a unital algebra A such as a C*-algebra or a von Neumann algebra. The algebra comes equipped with a noncommutative expectation, a linear functional φ: A → C such that φ(1) = 1. Unital subalgebras A1, ..., Am are then said to be freely independent if the expectation of the product a1...an is zero whenever each aj has zero expectation, lies in an Ak, no adjacent aj's come from the same subalgebra Ak, and n is nonzero. Random variables are freely independent if they generate freely independent unital subalgebras.

One of the goals of free probability (still unaccomplished) was to construct new invariants of von Neumann algebras and free dimension is regarded as a reasonable candidate for such an invariant. The main tool used for the construction of free dimension is free entropy.

The relation of free probability with random matrices is a key reason for the wide use of free probability in other subjects. Voiculescu introduced the concept of freeness around 1983 in an operator algebraic context; at the beginning there was no relation at all with random matrices. This connection was only revealed later in 1991 by Voiculescu; he was motivated by the fact that the limit distribution which he found in his free central limit theorem had appeared before in Wigner's semi-circle law in the random matrix context.

The free cumulant functional (introduced by Roland Speicher)[1] plays a major role in the theory. It is related to the lattice of noncrossing partitions of the set { 1, ..., n } in the same way in which the classic cumulant functional is related to the lattice of all partitions of that set.

## Contents

* 1 See also
* 2 References
* 2.1 Citations
* 2.2 Introductory surveys
* 2.3 Introductory monographs
* 2.4 Research monographs
* 3 External links

## See also

* Random matrix
* Wigner semicircle distribution
* Circular law
* Free convolution

## References

### Citations

1. ↑ Speicher, Roland (1994), "Multiplicative functions on the lattice of non-crossing partitions and free convolution", Mathematische Annalen 298 (4): 611–628, doi:10.1007/BF01459754 .

### Introductory surveys

* Survey articles of Roland Speicher on free probability, last updated in 1998.
* Biane, Philippe (1998-09-30). "Free probability for probabilists". arXiv:math/9809193.
* Mitchener, P. D. (2005). "Non-Commutative Probability Theory". http://www.mitchener.staff.shef.ac.uk/free.pdf.
* Tao, Terence (2010-02-10). "254A, Notes 5: Free probability". http://terrytao.wordpress.com/2010/02/10/245a-notes-5-free-probability/.

### Introductory monographs

* Nica, Alexandru; Speicher, Roland (2006). Lectures on the Combinatorics of Free Probability. London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511735127. ISBN 978-0-521-85852-6. https://www.math.uni-sb.de/ag/speicher/publikationen/Nica-Speicher.pdf.
* Voiculescu, Dan, ed (2016). Free probability and operator algebras. Münster lectures in mathematics. 1. Zürich: European Mathematical Society. doi:10.4171/165. ISBN 978-3-03719-165-1. https://ems.press/content/book-files/21394.
* Mingo, James A.; Speicher, Roland (2017) (in en). Free Probability and Random Matrices. Fields Institute Monographs. Springer. doi:10.1007/978-1-4939-6942-5. ISBN 978-1-4939-6942-5. https://rolandspeicher.com/wp-content/uploads/2019/02/mingo-speicher.pdf.
* Speicher, Roland (2019). "Free Probability Theory". https://rolandspeicher.com/wp-content/uploads/2025/04/free-probability_v2.pdf.
* Potters, Marc; Bouchaud, Jean-Philippe (2020-11-30). A First Course in Random Matrix Theory: for Physicists, Engineers and Data Scientists. Cambridge University Press. doi:10.1017/9781108768900. ISBN 978-1-108-76890-0.

### Research monographs

* Voiculescu, D. V.; Dykema, K. J.; Nica, A. (1992). Free random variables: a noncommutative probability approach to free products with applications to random matrices, operator algebras, and harmonic analysis on free groups. CRM monograph series. Providence, R.I., USA: American Mathematical Society. ISBN 978-0-8218-6999-4. OCLC 26852238.
* Voiculescu, Dan, ed (1997). Free probability theory. Fields Institute Communications (1 ed.). Providence, RI: American Mathematical Society. doi:10.1090/fic/012. ISBN 978-0-8218-0675-3.
* Speicher, Roland (1998) (in en). Combinatorial theory of the free product with amalgamation and operator-valued free probability theory. Memoirs of the American Mathematical Society. 132. doi:10.1090/memo/0627. https://rolandspeicher.com/wp-content/uploads/2020/05/memoirs.pdf.
* Hiai, Fumio; Petz, Dénes (2000). The semicircle law, free random variables, and entropy. Mathematical surveys and monographs. Providence, RI: American Mathematical Society. ISBN 978-0-8218-2081-0.

## External links

* Voiculescu receives NAS award in mathematics -- contains a readable description of free probability.
* RMTool -- A MATLAB-based free probability calculator.

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