Free probability

The free probability theory is a branch of mathematics, which was founded in 1985 by Dan Voiculescu. The theory arose from the search for a better understanding of certain algebras of operators on Hilbert spaces. Voiculescu isolated while the concept of " freeness " or "free independence" as essential structural and initiated the free probability theory as the study of this structure, detached from its concrete occurrence in operator algebras. A basic idea is to see the clear independence in analogy to the concept of independence of stochastic random variables and to develop the theory in this sense as a kind of probability theory for non- commuting variables. The discovery of Voiculescu (1991 ) that even large classes of random matrices are asymptotically free, marked the transition of the free probability theory from a hochspezialierten theory for certain operator algebras to a fundamental theory with wide application circuit. In particular, the free probability theory provides new methods for the calculation of eigenvalue distributions of random matrices, which are also of interest in applied fields such as are, for example, in wireless communications.

Notation: non- commutative probability space and random variable

A non- commutative probability space is a tuple consisting of a unitary algebra and a linear functional with.

You speak the elements in the algebra as generalized (or non-commutative ) random variables.

Definition: Freeness or free independence

In place of the stochastic independence occurs in the free probability theory, the concept of freeness or free independence, which is defined as follows:

Be an arbitrary index set.

1) Be a family of unitary subalgebras of. Then the names of the free ( or freely independent) if the following holds:

For all and all with the index up and running and in addition must apply. This means that adjacent elements do not originate from the same sub algebra and that the elements are each centered.

2) random variables for hot free, if their production of unitary subalgebras are free.

3) If you have a sequence of subalgebras or random variable, and the above relations are valid only asymptotically, we speak of free asymptotic independence.

Freeness as a rule for calculating mixed moments

It now proves easily by induction the following fundamental observation: Are the free subalgebras with respect to, the values ​​of on the generated by the algebra are uniquely determined by the values ​​of all constraints of the and by the freeness condition. In this sense, the mixed moments of free random variables by the moments of the individual random variables are determined. If and are free so you have for example and that

These examples show that the free independence as classical independence can be considered as a rule for the calculation of joint torques; However, this rule is different than the classic. The free independence is thus to be seen analogous to the classical independence, but it is not a generalization of it. In particular, classical random variable can only be free if at least one of the random variables is constant. The free independence is an intrinsically non- commutative concept.

The free central limit theorem

Be a non- commutative probability space and a sequence of identically distributed and independent random variables with mean and variance. Then converges

In distribution to a semi-circle element, that is, for all natural applies:

.

More free stochastic results

The free central limit theorem is only an example of a very rich free probability theory, parallel to the classical probability theory. So you have the binary operation of the free convolution on the real probability measures, this corresponds to the addition of free random variables. The R- transform is the equivalent of the logarithm of the Fourier transform and allows a systematic and effective computation of the free convolution of probability measures. The corresponding multiplicative versions are given by the multiplicative free fold ( which corresponds to the product of free random variables ), and the S transform.

The coefficients of the R transformation, the so-called free cumulants have a combinatorial interpretation in non- crossing partitions. The latter allows a combinatorial approach to free probability theory.

Connection with random matrices

The fact that the semi-circle distribution emerges not only as a limit in the free central limit theorem, but also in Wigner's semicircle law as the asymptotic distribution of the eigenvalues ​​of Gaussian random matrices, pointing to a deeper connection between free probability theory and random matrices. Voiculescu pursued further this connection and was able to show in 1991 that free independence for large classes of random matrices occurs asymptotically:

Are two series of matrices and in the non- commutative probability space, each with existing threshold distribution, ie for all natural limits exist and Be a unitary Zufallsmatriz, distributed according to the normalized Hair Mass on the unitary matrices. Then, almost surely asymptotically free.