Free probability
Encyclopedia
Free probability is a mathematical
theory that studies non-commutative random variable
s. The "freeness" or free independence
property is the analogue of the classical notion of independence
, and it is connected with free product
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
s. 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 if 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 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-star 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, and no adjacent aj's come from the same subalgebra Ak. 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
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 free cumulant functional (introduced by Roland Speicher) 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 partitions
of that set.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
theory that studies non-commutative random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
s. The "freeness" or free independence
Free independence
In the mathematical theory of free probability, the notion of free independence was introduced by Dan Voiculescu. The definition of free independence is parallel to the classical definition of independence, except that the role of Cartesian products of measure spaces is played by the notion of a...
property is the analogue of the classical notion of independence
Statistical independence
In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs...
, and it is connected with free product
Free product
In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties...
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
Operator algebra
In functional analysis, an operator algebra is an algebra of continuous linear operators on a topological vector space with the multiplication given by the composition of mappings...
s. Given a free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
on some number of generators, we can consider the von Neumann algebra
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. They were originally introduced by John von Neumann, motivated by his study of single operators, group...
generated by the group algebra
Group algebra
In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra , such that representations of the algebra are related to representations of the group...
, which is a type II1 factor. The isomorphism problem asks if 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
Random matrix
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable. Many important properties of physical systems can be represented mathematically as matrix problems...
, combinatorics
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
, representations
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
of symmetric group
Symmetric group
In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
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-star algebra or a von Neumann algebra
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. They were originally introduced by John von Neumann, motivated by his study of single operators, group...
. The algebra comes equipped with a noncommutative expectation, a linear
Linear
In mathematics, a linear map or function f is a function which satisfies the following two properties:* Additivity : f = f + f...
functional
Functional
Generally, functional refers to something able to fulfill its purpose or function.*Functionalism and Functional form, movements in architectural design*Functional group, certain atomic combinations that occur in various molecules, e.g...
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, and no adjacent aj's come from the same subalgebra Ak. 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
Invariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
of von Neumann algebra
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. They were originally introduced by John von Neumann, motivated by his study of single operators, group...
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 free cumulant functional (introduced by Roland Speicher) plays a major role in the theory. It is related to the lattice of noncrossing partition
Noncrossing partition
In combinatorial mathematics, the topic of noncrossing partitions has assumed some importance because of its application to the theory of free probability...
s of the set { 1, ..., n } in the same way in which the classic cumulant functional is related to the lattice of all partitions
Partition of a set
In mathematics, a partition of a set X is a division of X into non-overlapping and non-empty "parts" or "blocks" or "cells" that cover all of X...
of that set.
See also
- Random matrixRandom matrixIn probability theory and mathematical physics, a random matrix is a matrix-valued random variable. Many important properties of physical systems can be represented mathematically as matrix problems...
- Wigner semicircle distribution
- Circular lawCircular lawIn probability theory, more specifically the study of random matrices, the circular law describes the distribution of eigenvalues of an n \times n random matrix with independent and identically distributed entries in the limit n \to \infty ....
- Free deconvolutionFree deconvolutionFree convolution is the free probability analog of the classical notion of convolution of probability measures. Due to the non-commutative nature of free probability theory, one has to talk separately about additive and multiplicative free convolution, which arise from addition and multiplication...
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 Radio Applications of Free Probability to Wireless Communications.
- survey articles of Roland Speicher on free probability.