Free probability is a

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

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. Most familiar as the name of ...

random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...

s. The "freeness" or

free independence In the mathematical theory of free probability, the notion of free independence was introduced by Dan Voiculescu.D. Voiculescu, K. Dykema, A. Nica, "Free Random Variables", CIRM Monograph Series, AMS, Providence, RI, 1992 The definition of free in ...

property is the analogue of the classical notion of

independence Independence is a condition of a person, 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 stat ...

, 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, and is ...

s. 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 of ...

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''−1' ...

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 algebra ...

generated by the group algebra, which is a type II₁

factor Factor, a Latin word meaning "who/which acts", 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, suc ...

. The isomorphism problem asks whether these are

isomorphic In mathematics, an isomorphism is a structure-preserving mapping 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 them. The word i ...

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 In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can be represented mathemat ...

combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many a ...

representations ''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It ...

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 group ...

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 quantum state, 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 re ...

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 continuou ...

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 map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the ...

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 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 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 Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of ...

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 Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes ...

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.
blog on free probability

{{DEFAULTSORT:Free Probability Functional analysis Exotic probabilities

See also

References

Citations

Sources

External links