Editing Free probability

{{Short description|Mathematical theory on random variables}}
'''Free probability''' is a [[mathematics|mathematical]] theory that studies [[non-commutative]] [[random variable]]s. The "freeness" or [[free independence]] property is the analogue of the classical notion of [[statistical independence|independence]], and it is connected with [[free product]]s.
This theory was initiated by [[Dan Voiculescu (mathematician)|Dan Voiculescu]] around 1986 in order to attack the free group factors isomorphism problem, an important unsolved problem in the theory of [[operator algebra]]s. Given a [[free group]] on some number of generators, we can consider the [[von Neumann algebra]] generated by the [[group algebra of a locally compact group|group algebra]], which is a type II<sub>1</sub> [[von Neumann algebra#Factors|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|random matrix theory]], [[combinatorics]], [[group representation|representations]] of [[symmetric group]]s, [[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 ''A''<sub>1</sub>, ..., ''A''<sub>''m''</sub> are then said to be '''freely independent''' if the expectation of the product ''a''<sub>1</sub>...''a''<sub>''n''</sub> is zero whenever each ''a''<sub>''j''</sub> has zero expectation, lies in an ''A''<sub>''k''</sub>, no adjacent ''a''<sub>''j''</sub>'s come from the same subalgebra ''A''<sub>''k''</sub>, 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 [[invariant (mathematics)|invariants]] of [[von Neumann algebra]]s 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 [[Cumulant#Free cumulants|free cumulant]] functional (introduced by [[Roland Speicher]])<ref name="bnt-s">{{citation
 | last = Speicher | first = Roland
 | doi = 10.1007/BF01459754
 | issue = 4
 | journal = Mathematische Annalen
 | mr = 1268597
 | pages = 611–628
 | title = Multiplicative functions on the lattice of non-crossing partitions and free convolution
 | volume = 298
 | year = 1994}}.</ref> plays a major role in the theory.  It is related to the lattice of [[noncrossing partition]]s of the set { 1, ..., ''n'' } in the same way in which the classic cumulant functional is related to the lattice of ''all'' [[partition of a set|partitions]] of that set.

== See also ==
* [[Random matrix]]
* [[Wigner semicircle distribution]]
* [[Circular law]]
* [[Free convolution]]

== References ==
=== Citations ===
{{Reflist}}

=== Sources ===
{{refbegin}}
* D.-V. Voiculescu, N. Stammeier, M. Weber (eds.): [http://www.ems-ph.org/books/book.php?proj_nr=208''Free Probability and Operator Algebras''], Münster Lectures in Mathematics, EMS, 2016
* James A. Mingo, Roland Speicher: [https://www.springer.com/de/book/9781493969418''Free Probability and Random Matrices'']. Fields Institute Monographs, Vol. 35, Springer, New York, 2017.
* A. Nica, R. Speicher: [http://www.cambridge.org/us/catalogue/catalogue.asp?isbn=0521858526 ''Lectures on the Combinatorics of Free Probability.''] Cambridge University Press, 2006, {{ISBN|0-521-85852-6}}
* Fumio Hiai and Denis Petz, ''The Semicircle Law, Free Random Variables, and Entropy'', {{ISBN|0-8218-2081-8}}
* Mitchener, P.D. (2005) [http://www.mitchener.staff.shef.ac.uk/free.pdf ''Non-Commutative Probability Theory''], preprint
* Voiculescu, D. V.; Dykema, K. J.; Nica, A. ''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, 1. American Mathematical Society, Providence, RI, 1992. {{ISBN|0-8218-6999-X}}
* [[Terence Tao]], [http://terrytao.wordpress.com/2010/02/10/245a-notes-5-free-probability/ 254A, Notes 5: Free probability] (10 February, 2010), course notes for graduate course on "Topics in random matrix theory"
* Roland Speicher: [https://rolandspeicher.com/wp-content/uploads/2025/04/free-probability_v2.pdf Free Probability Theory], course notes
{{refend}}

== External links ==
* [https://www.ams.org/notices/200405/comm-nas.pdf Voiculescu receives NAS award in mathematics] &mdash; contains a readable description of free probability.
* [https://www.mit.edu/~raj/rmtool RMTool] &mdash; A MATLAB-based free probability calculator.
* Alcatel-Lucent Chair on Flexible Radio [http://www.supelec.fr/d2ri/flexibleradio/cours/deconvolution.pdf Applications of Free Probability to Wireless Communications].
* [https://www.math.uni-sb.de/ag/speicher/speicher_publikationenE.html survey articles] of Roland Speicher on free probability.

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[[Category:Functional analysis]]
[[Category:Exotic probabilities]]
[[Category:Free probability theory| ]]