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.

Typically the random variables lie in a unital algebra such as a C-star algebra or a von Neumann algebra. The algebra comes equipped with an expectation. 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.

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 .

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.

See also

• Random matrix
• Wigner semicircle distribution

External link

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