Free Probability
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
A
1
, ...,
A
n
are then said to be
free
if the expectation of the product
a
1
...
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
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