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Wikipedia / Free probability

Free probability is a mathematical theory that studies non-commutative random variables. The "freeness" or free independence 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, an important unsolved problem in the theory of operator algebras. 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 whether 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 groups, large deviations, quantum information theory and other theories were established. Free probability is currently undergoing active research.

Noncommutative random variables and freeness

Free random variables

a until algebra, such as a C*-algebra or a von Neumann algebra.

times the usual matrix trace.

Definition of freeness

-algebras). Then we say that the algebras are free or freely independent has zero expectation and belongs to one of the subalgebras and no two consecutive elements belong to the same subalgebra.

-freeness.)

Asymptotic freeness

As mentioned above, one of the key reasons for interest in free probability is its connection to random matrix theory. This connection comes from Voiculescu's asymptotic freeness of them are invariant in law under conjugation by arbitrary unitary matrices, then these random matrices are asymptotically free in a tracial von Neumann algebra.

in terms of the limiting eigenvalues distributions of the two matrices separately, using an operation known as the free additive convolution.

Law of a normal random variable

. Then the spectral theorem. We then define the law is the unique measure supported on the spectrum.

-th Catalan number will be the standard Wigner semicircle distribution.

is the empirical spectral measure.

History of the subject

One of the goals of free probability (still unaccomplished) was to construct new invariants 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 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 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

References

Citations

Introductory surveys

Introductory monographs

Research monographs