Free probability is a
mathematical
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
theory that studies
non-commutative random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
s. The "freeness" or
free independence property is the analogue of the classical notion of
independence
Independence is a condition of a nation, country, or state, in which residents and population, or some portion thereof, exercise self-government, and usually sovereignty, over its territory. The opposite of independence is the status of ...
, 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 algebra
In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with the multiplication given by the composition of mappings.
The results obtained in the study o ...
s. Given a
free group on some number of generators, we can consider the
von Neumann algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra.
Von Neumann al ...
generated by the
group algebra, which is a type II
1 factor. The isomorphism problem asks whether these are
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
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
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
,
representations of
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
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
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra.
Von Neumann al ...
. The algebra comes equipped with a noncommutative expectation, a
linear functional φ: ''A'' → C such that φ(1) = 1. Unital subalgebras ''A''
1, ..., ''A''
''m'' are then said to be freely independent if the expectation of the product ''a''
1...''a''
''n'' is zero whenever each ''a''
''j'' has zero expectation, lies in an ''A''
''k'', no adjacent ''a''
''j'''s come from the same subalgebra ''A''
''k'', 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 algebra
In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra.
Von Neumann al ...
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
free cumulant functional (introduced by
Roland Speicher)
[.] plays a major role in the theory. It is related to the lattice of
noncrossing partitions of the set in the same way in which the classic cumulant functional is related to the lattice of ''all''
partitions of that set.
See also
*
Random matrix
*
Wigner semicircle distribution
*
Circular law
*
Free convolution
References
Citations
Sources
* D.-V. Voiculescu, N. Stammeier, M. Weber (eds.)
Probability and Operator Algebras'' Münster Lectures in Mathematics, EMS, 2016
* James A. Mingo, Roland Speicher
Probability and Random Matrices'' Fields Institute Monographs, Vol. 35, Springer, New York, 2017.
* A. Nica, R. Speicher
''Lectures on the Combinatorics of Free Probability.''Cambridge University Press, 2006,
* Fumio Hiai and Denis Petz, ''The Semicircle Law, Free Random Variables, and Entropy'',
* Mitchener, P.D. (2005
''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.
*
Terence Tao254A, Notes 5: Free probability(10 February, 2010), course notes for graduate course on "Topics in random matrix theory"
* Roland Speicher
Free Probability Theory course notes
External links
Voiculescu receives NAS award in mathematics— contains a readable description of free probability.
RMTool— A MATLAB-based free probability calculator.
* Alcatel-Lucent Chair on Flexible Radi
Applications of Free Probability to Wireless Communications
of Roland Speicher on free probability.
{{DEFAULTSORT:Free Probability
Functional analysis
Exotic probabilities