Free probability theory

The free probability theory is a branch of mathematics that in 1985 by Dan Voiculescuwas established. The theory arose from the search for a better understanding of certain algebras of operators on Hilbert spaces. Voiculescu isolated the concept of “freeness” or “free independence” as an essential structure and initiated free probability theory as the investigation of this structure, detached from its concrete occurrence in operator algebras. A basic idea is to see the free independence in analogy to the concept of independence from 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 become asymptotically free, marked the transition of free probability theory from a specialized theory for certain operator algebras to a fundamental theory with a wide range of applications. In particular, free probability theory provides new methods for calculating eigenvalue distributions of random matrices, which are also of interest in applied areas, such as B. wireless communication.

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 . ${\displaystyle (A,\varphi )}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle A}$ ${\displaystyle \varphi :A\rightarrow \mathbb {C} }$ ${\displaystyle \varphi (1)=1}$

The elements in algebra are referred to as generalized (or non-commutative) random variables . ${\displaystyle A}$

Definition: Freeness or free independence

In free probability theory, the stochastic independence is replaced by the concept of freeness or free independence, which is defined as follows:

Let be any index set, then: ${\displaystyle I}$

1) Let be a family of unitary subalgebras of . Then they are called free (or free independent) if: ${\displaystyle \left(A_{i}\right)_{i\in I}}$ ${\displaystyle A}$ ${\displaystyle A_{i}}$

${\displaystyle \varphi (a_{1}a_{2}\ldots a_{k})=0}$

for everyone and everyone , whereby the index runs from to and additionally and must apply. This means that neighboring elements do not come from the same sub-algebra and that the elements are each centered. ${\displaystyle k\in \mathbb {N} }$ ${\displaystyle a_{j}\in A_{i(j)}}$ ${\displaystyle j}$ ${\displaystyle 1}$ ${\displaystyle k}$ ${\displaystyle \varphi (a_{j})=0}$ ${\displaystyle i(1)\neq i(2)\neq \ldots \neq i(k)}$

2) Random variables for are called free, if the unitary subalgebras they generate are free. ${\displaystyle b_{i}\in A}$ ${\displaystyle i\in I}$

3) If one has a sequence of sub-algebras or random variables and the above relations only apply asymptotically, one speaks of asymptotic free independence.

Freeness as a rule for calculating mixed moments

The following fundamental observation can now easily be proven by induction: If the subalgebras are free with respect to , then the values ​​of on are uniquely determined by the values ​​of all restrictions on and by the freeness condition. In this sense, the mixed moments of free random variables are determined by the moments of the individual random variables. If and are free, you have z. B. for and that ${\displaystyle A_{i}}$ ${\displaystyle \varphi }$ ${\displaystyle \varphi }$ ${\displaystyle A_{i}}$ ${\displaystyle \varphi }$ ${\displaystyle A_{i}}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle a_{1},a_{2}\in A}$ ${\displaystyle b,b_{1},b_{2}\in B}$

${\displaystyle \varphi (ab)=\varphi (a)\varphi (b)}$

${\displaystyle \varphi (a_{1}ba_{2})=\varphi (a_{1}a_{2})\varphi (b)}$

${\displaystyle \varphi (a_{1}b_{1}a_{2}b_{2})=\varphi (a_{1}a_{2})\varphi (b_{1})\varphi (b_{2})+\varphi (a_{1})\varphi (a_{2})\varphi (b_{1}b_{2})-\varphi (a_{1})\varphi (b_{1})\varphi (a_{2})\varphi (b_{2})}$

These examples show that free independence, like classical independence, can be viewed as a rule for calculating mixed moments; however, this rule is different from the classic one. Free independence is to be seen analogously to classical independence, but it is not a generalization of it. In particular, classic random variables can only be free if at least one of the random variables is constant. Free independence is an intrinsically non-commutative concept.

The free central limit theorem

Let be a non-commutative probability space and a sequence of identically distributed and free random variables with mean and variance . Then converges ${\displaystyle (A,\varphi )}$ ${\displaystyle a_{1},a_{2},\dotsc \in A}$ ${\displaystyle \varphi (a_{i})=0}$ ${\displaystyle \varphi (a_{i}^{2})=1}$

${\displaystyle S_{N}:={\frac {a_{1}+\cdots +a_{N}}{\sqrt {N}}}}$

in distribution against a semicircular element, d. H. for all natural numbers the following applies: ${\displaystyle n}$

${\displaystyle \lim _{N\to \infty }\varphi (S_{N}^{n})={\frac {1}{2\pi }}\int _{-2}^{+2}t^{n}{\sqrt {4-t^{2}}}dt}$.

More free stochastic results

The free central limit theorem is just one example of a very rich free probability theory , parallel to classical probability theory. So one has the binary operation of free convolution on the real probability measures; this corresponds to the addition of free random variables. The R transformation is the equivalent of the logarithm of the Fourier transformation and allows a systematic and effective calculation of the free convolution of probability measures. The corresponding multiplicative versions are given by the multiplicative free convolution (which corresponds to the product of free random variables) and the S transformation . ${\displaystyle \boxplus }$ ${\displaystyle \boxtimes }$

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

Relation to random matrices

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

Let and be two sequences of matrices in the non-commutative probability spaces with an existing limit value distribution, i.e. H. for all natural numbers the limit values ​​exist and Let be a unitary random matrix, distributed according to the normalized Haar measure on the unitary matrices. Then, and almost certainly asymptotically free. ${\displaystyle A_{N}}$ ${\displaystyle B_{N}}$ ${\displaystyle N\times N}$ ${\displaystyle (M_{N}(\mathbb {C} ),tr_{N})}$ ${\displaystyle n}$ ${\displaystyle \lim _{N\to \infty }\varphi (A_{N}^{n})}$ ${\displaystyle \lim _{N\to \infty }\varphi (B_{N}^{n}).}$ ${\displaystyle U_{N}}$ ${\displaystyle A_{N}}$ ${\displaystyle U_{N}B_{N}U_{N}^{*}}$

Relation to operator algebras

The basic example of free independence appears in connection with Von Neumann algebras of free products of groups .

Be the free product of groups . Let be the associated group von Neumann algebra and the associated trace state that corresponds to the neutral element of the group. Then can be identified with a sub-algebra of , and with regard to this all of these are free. This free independence is nothing else than the paraphrase with the help of the fact that the groups are free as subgroups . The definition of free independence was modeled after this example. That free in free independence refers to it. ${\displaystyle G=\prod _{i}G_{i}}$ ${\displaystyle G_{i}}$ ${\displaystyle L(G)}$ ${\displaystyle \varphi }$ ${\displaystyle L(G_{i})}$ ${\displaystyle L(G)}$ ${\displaystyle \varphi }$ ${\displaystyle L(G_{i})}$ ${\displaystyle \varphi }$ ${\displaystyle G_{i}}$ ${\displaystyle G}$

An important special case of the von Neumann group algebras are the free group factors , whereby the non-commutative free group is with generators. Voiculescu's motivation was the free group isomorphism problem : Are the Von Neumann algebras and for isomorphic? ${\displaystyle L(\mathbb {F} _{n})}$ ${\displaystyle \mathbb {F_{n}} }$ ${\displaystyle n}$ ${\displaystyle L(\mathbb {F} _{n})}$ ${\displaystyle L(\mathbb {F} _{m})}$ ${\displaystyle n,m\geq 2;n\not =m}$

This isomorphism problem is still open, but free probability theory has made significant advances in our understanding of free group factors. In this way, Dykema and Radulescu, building on previous work by Voiculescu, were able to show that they are either all isomorphic or are different in pairs. ${\displaystyle L(\mathbb {F} _{n})}$

Free entropy and operator algebras

An important direction in free probability theory in recent years is free entropy theory. The free entropy of a tuple of operators is a measure of how many tuples of matrices there are that approximate the tuple under consideration in distribution. An important application of free entropy was Voiculescu's proof that the free group factors do not have any Cartan subalgebras - this provided a counterexample to the conjecture that has been open since the 1960s that all II factors have such a subalgebra. ${\displaystyle _{1}}$

literature

• D.-V. Voiculescu, N. Stammeier, M. Weber (eds.): Free Probability and Operator Algebras , Münster Lectures in Mathematics, EMS, 2016.
• James A. Mingo, Roland Speicher: Free Probability and Random Matrices . Fields Institute Monographs, Vol. 35, Springer Verlag, New York, 2017.
• Alexandru Nica, Roland Speicher: Lectures on the Combinatorics of Free Probability (= London Mathematical Society Lecture Note Series. Vol. 335). Cambridge University Press, Cambridge et al. 2006, ISBN 0-521-85852-6 .
• Fumio Hiai, Dénes Petz: The Semicircle Law, Free Random Variables, and Entropy (= Mathematical Surveys and Monographs. Vol. 77). American Mathematical Society, Providence RI 2000, ISBN 0-8218-2081-8 .
• DV Voiculescu, KJ Dykema, A. Nica: Free random variables. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups (= CRM Monograph Series. Vol. 1). American Mathematical Society, Providence RI 1992, ISBN 0-8218-6999-X .
• Roland Speicher: Free Probability Theory , lecture manuscript.

Web links

• Voiculescu receives NAS award in mathematics (PDF file; 45 kB)
• RMTool - A MATLAB based free probability calculator.
• Review article by Roland Speicher
• Recording of a series of lectures on free probability theory (in English)
• Roland Speicher's blog on free probability theory