Free probability is a mathematical theory which studies non-commutative random variables. The "freeness" property is the analogue of the classical notion of independence, 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 (which still remains open) but later connections to the random matrix theory, combinatorics, representations of symmetric groups, large deviations and other theories were established. Free probability is currently undergoing active research. Main article: History of mathematics The evolution of mathematics can be seen to be an ever increasing series of abstractions. ... In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ... In abstract algebra, the free product of groups constructs a group from two or more given ones. ... 1986 is a common year starting on Wednesday of the Gregorian calendar. ... Free group factors isomorphism problem is an unsolved mathematical question. ... In probability theory and statistics, a random matrix is a matrix-valued random variable. ... Combinatorics is a branch of mathematics that studies finite collections of objects that satisfy specified criteria. ... Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ...

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, ..., A_n are then said to be free if the expectation of the product a₁...a_n is zero whenever each a_j has zero expectation, lies in an A_k and no adjacent a_j's come from the same subalgebra A_k. Random variables are free if they generate free unital subalgebras. In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ... Algebra is a branch of mathematics which studies structure and quantity. ... C*-algebras are an important area of research in functional analysis. ... A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space which is closed in the weak operator topology, or equivalently, in the strong operator topology (under pointwise convergence) and contains the identity operator. ... The word linear comes from the Latin word linearis, which means created by lines. ... Generally, functional refers to something with and able to fulfill its purpose or function. ...

One of the goals of free probability (still unaccomplished) was to construct new invariant of von Neumann algebras 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. An invariant in mathematics is something that does not change under a set of transformations. ... A von Neumann algebra is a *-algebra of bounded operators on a Hilbert space which is closed in the weak operator topology, or equivalently, in the strong operator topology (under pointwise convergence) and contains the identity operator. ...

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 { 1, ..., n } in the same way in which the classic cumulant functional is related to the lattice of all partitions of that set. Cumulants of probability distributions In probability theory and statistics, the cumulants κn of a probability distribution are given by where X is any random variable whose probability distribution is the one whose cumulants are taken. ... In combinatorial mathematics, the topic of noncrossing partitions has assumed some importance because of (among other things) its application to the theory free probability. ... A partition of U into 6 blocks: a Venn diagram representation. ...

See also

• Random matrix
• Wigner semicircle distribution

In probability theory and statistics, a random matrix is a matrix-valued random variable. ... The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution supported on the interval [−R, R] the graph of whose probability density function f is a semicircle of radius R centered at (0, 0) and then suitably normalized (so that it is really a semi-ellipse...

External link

• Voiculescu receives NAS award in mathematics - containing a readable description of free probability