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 In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...

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 product In mathematics, specifically group theory, the free product is an operation that takes two groups ''G'' and ''H'' and constructs a new The result contains both ''G'' and ''H'' as subgroups, is generated by the elements of these subgroups, an ...

s. This theory was initiated by

Dan Voiculescu Dan Voiculescu (; born September 25, 1946), also known as ''"Varanul"'', lit. 'the Monitor lizard', or "Felix Voiculescu", is a Romanian politician and businessman. He is the founder and former president of the Romanian Humanist Party (PUR), la ...

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 In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''− ...

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 II₁

factor Factor (Latin, ) may refer to: Commerce * Factor (agent), a person who acts for, notably a mercantile and colonial agent * Factor (Scotland), a person or firm managing a Scottish estate * Factors of production, such a factor is a resource used ...

. 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 ...

large deviations In probability theory, the theory of large deviations concerns the asymptotic behaviour of remote tails of sequences of probability distributions. While some basic ideas of the theory can be traced to Laplace, the formalization started with insura ...

quantum information theory Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both t ...

and other theories were established. Free probability is currently undergoing active research. Typically the random variables lie in a

unital algebra In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...

''A'' such as a

C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of contin ...

or a

. The algebra comes equipped with a noncommutative expectation, a

linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of ...

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

s and

free dimension Free may refer to: Concept * Freedom, the ability to act or change without constraint or restriction * Emancipate, attaining civil and political rights or equality * Free (''gratis''), free of charge * Gratis versus libre, the difference betw ...

is regarded as a reasonable candidate for such an invariant. The main tool used for the construction of

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 Roland Speicher (born 12 June 1960) is a German mathematician, known for his work on free probability theory. He is a professor at the Saarland University. After winning the 1979 German national competition Jugend forscht in the field of mathem ...

). plays a major role in the theory. It is related to the lattice of

noncrossing partition In combinatorial mathematics, the topic of noncrossing partitions has assumed some importance because of (among other things) its application to the theory of free probability. The number of noncrossing partitions of a set of ''n'' elements is t ...

s of the set in the same way in which the classic cumulant functional is related to the lattice of ''all'' partitions of that set.

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 Tao
254A, 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

See also

References

Citations

Sources

External links